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A TREATISE 


ON 

PHOTOGRAPHIC  OPTICS 


/ 


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Full  size. 


Divergenf. 


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Full  size. 


ConvergenI' . 


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ijnvergenl’ . 


A TREATISE 


ON 

PHOTOGRAPHIC  OPTICS 


BY 

R.  S.  COLE,  M.A. 

LATE  SCHOLAR  OF  EMMANUEL  COLLEGE,  CAMBRIDGE, 
ASSISTANT-MASTER,  MARLBOROUGH  COLLEGE 


ILLUSTRATED 


NEW  YORK 

D.  VAN  NOSTRAND  COMPANY 


S 25  5-.  <5-5 

■PEMOTE  STORAGE 

PREFACE 


The  object  of  this  treatise  is  to  provide  an  account 
of  the  principles  of  Optics,  so  far  as  they  apply  to 
Photography,  in  a form  which  is  of  scientific  value, 
while  not  of  too  abstruse  a nature  to  place  it  beyond 
the  reach  of  all  but  the  professional  mathematician  or 
physicist. 

I have  attempted  to  steer  a middle  course  between 
giving  too  much  mathematics  and  giving  none  at  all  ; 
the  former  course  would  restrict  the  book  to  a few, 
while  the  latter  would  deprive  it  of  all  real  value. 

To  make  the  mathematics  as  intelligible  as  possible, 
most  of  the  results  have  been  illustrated  by  worked 
numerical  examples,  and  symbolical  results  have  been 
expressed  in  words. 

The  chapter  on  aberration  necessarily  contains  a 
<n  certain  amount  of  algebraical  formuhe,  too  many  per- 
^ haps  for  many  readers,  but  care  has  been  taken  to 
^ explain  clearly,  by  means  of  diagrams,  the  principles 
which  underlie  the  formulae. 

I have  to  thank  Captain  Abney  for  giving  me  per- 
mission to  use  many  of  his  diagrams.  Major  Darwin  for 
allowing  me  to  quote  from  his  paper  on  lens  testing 
at  Kew  Observatory,  and  the  publishers  of  Photography 
Annual  for  allowing  me  to  copy  the  tables  contained 
in  §1  127,  151,  and  156  ; also  I owe  it  to  the  kindness 
of  the  Sand  ell  Plate  Company  that  I am  able  to  give 
an  example  of  a “ test  film  ” for  the  speed  of  plates. 

My  thanks  are  also  due  to  the  Rev.  W.  F.  H.  Curtis, 
who  first  suggested  to  me  to  write,  and  has  given  me 
many  valuable  suggestions  and  much  help. 

Marlborough, 

October  1898, 


CONTENTS 

CHAP.  PAGE 

INTRODUCTION  ......  1 

I.  ON  LIGHT  ......  3 

II.  ELEMENTARY  THEORY  OF  LENSES  . . 39 

III.  ABERRATION  . . . . . .122 

IV.  THE  CORRECTION  OF  ABERRATION  AND  THE 

DESIGN  OF  LENSES  . . . .190 

* 

V.  LENS  TESTING  . . . . . .205 

VI.  EXPOSURE,  STOPS,  AND  SHUTTERS  . .251 

VII.  ENLARGEMENT,  REDUCTION,  DEPTH  OF  FOCUS, 

AND  HALATION  . . . . .294 

INDEX  .......  327 


t 


PHOTOGRAPHIC  OPTICS 


INTRODUCTION 

Many  books  have  been  written  on  optics,  and  of 
these  not  a few  are  designed  to  meet  the  wants  of 
photographers ; but  they  seem  to  have  done  little  as 
yet  to  drive  away  the  clouds  of  mystery  which  hang 
about  a lens.  Very  few  people  who  practise  photo- 
graphy know  much  about  the  instrument  which  they 
use. 

The  reason  of  this  ignorance  and  indifference  is  not 
hard  to  find ; it  arises  from  three  causes  : First,  the 
photographer  is  dazzled  by  the  results  of  his  work,  and 
his  main  idea  is  to  turn  out  each  picture  more  perfect 
than  the  preceding  one.  Secondly,  writers  on  optics 
have  for  some  occult  reason  deemed  it  necessary  to 
surround  the  simplest  optical  matters  with  a cloud  of 
symbols  repulsive  to  those  who  are  not  mathematically 
minded,  or  to  leave  out  all  the  mathematics,  and  with 
it  all  definiteness  and  accuracy  of  expression,  which 
causes  even  worse  confusion  than  symbols.  Thirdly,  a 
lens  is  far  too  complicated  to  be  made  by  an  amateur ; 
the  photographer  buys  his  lens  ready  for  use,  and  it  is 
nearly  impossible  to  find  out  its  construction,  and  so  as 
long  as  it  works  well  no  questions  are  asked. 

But  there  are  some  photographers  who  are  not  so 
much  engrossed  in  picture-making  that  they  can  attend 


B 


2 


INTKODUCTION 


to  nothing  else,  but  they  take  an  interest  in  all  matters 
connected  with  photography — the  chemistry,  optics, 
and  mechanics  of  it. 

It  is  for  such  photographers  that  this  book  is  written, 
and  it  is  intended  as  an  attempt  to  place  the  main 
ideas  of  optics  within  the  reach  of  those  who  care  to 
acquire  them,  without  an  undue  amount  of  mathematics 
or  sacrifice  of  precision. 

The  reader  is  presumed  to  possess  some  elementary 
mathematical  knowledge,  Euclid,  Algebra,  and  the 
simplest  parts  of  Trigonometry ; the  use  of  Trigonometry 
might  possibly  have  been  avoided,  but  it  would  have 
led  to  a great  deal  of  circumlocution.  The  mathe- 
matical working  will  be  kept  within  the  narrowest 
possible  limits,  consistent  with  giving  a satisfactory 
account ; to  have  reduced  it  further  would  have  been  to 
sacrifice  some  of  the  most  useful  results. 

Those  who  experience  a difficulty  in  reading  through 
a particular  piece  of  work  will  find  it  a great  assistance 
to  use  a pencil  and  piece  of  paper  as  they  read,  roughly 
jotting  down  and  working  out  calculations,  and  sketch- 
ing the  diagrams.  Besides  this,  the  various  experiments 
described  should,  when  possible,  be  performed  ; in  many 
cases  a lens,  a few  pieces  of  cardboard,  and  a candle  or 
two  will  be  all  that  is  required. 


CHAPTER  I 


ON  LIGHT 

1.  Since  photography  is  the  outcome  of  the  chemical 
action  of  light,  a thorough  knowledge  of  the  subject 
cannot  be  gained  without  some  acquaintance  with  the 
theories  which  have  been  proposed  to  account  for 
optical  phenomena,  and  specially  with  the  wave  theory, 
which  is  now  generally  considered  to  be  the  one  nearest 
to  the  truth.  Detailed  explanations  would,  of  course, 
be  out  of  place  here,  and  must  be  looked  for  in  special 
treatises,  but  a rapid  sketch  will  serve  to  show  what  is 
now  supposed  to  be  the  nature  of  the  machinery  at 
work,  and  to  explain  many  points  which  would  other- 
wise remain  obscure. 

2.  It  will  be  well  at  the  outset  to  get  a clear  idea 
of  the  problem  with  which  we  have  to  deal  when  we 
inquire  as  to  the  nature  of  light,  for  all  our  actions  are 
so  dependent  on  sight  that  it  is  not  easy  to  grasp  what 
it  is  that  needs  explanation.  Sight  has  been  called 
that  sense  which  compensates  for  the  want  of  ubiquity 
by  giving  information  about  objects  at  a distance ; this 
information  is  brought  to  us  through  the  agency  of  light. 

We  may  then  describe  light  as  some  kind  of  informa- 
tion which  is  conveyed  to  us  from  illuminated  bodies ; 
we  are  convinced  by  experience  that  it  is  not  merely  a 
subjective  phenomenon,  but  that,  in  the  space  between 
us  and  the  distant  object,  something  does  actually  take 
place. 

The  problem,  then,  is  to  discover  in  what  manner 


4 


PHOTOGRAPHIC  OPTICS 


this  information  is  conveyed  : to  do  this  we  must  take 
into  account  the  known  properties  of  light.  The  most 
important  properties  are  as  follows : — Light  is  pro- 
pagated through  air  and  homogeneous  media  of  uniform 
density  in  straight  lines  : it  is  reflected  if  it  meets  a 
smooth  opaque  body  : it  is  bent  or  refracted,  and  also 
refle(?ted,  when  it  passes  from  one  transparent  medium 
to  another,  e.  g.  from  glass  to  water : it  is  propagated 
through  air  or  space  with  the  definite,  though  enormous, 
velocity  of  188,000  miles  per  second — this  fact  was 
discovered  by  Hoemer  from  the  study  of  the  times  of 
eclipse  of  Jupiter’s  satellites,  and  has  since  been  amply 
verified  by  the  experiments  of  Fizeau  and  others. 

Let  us  consider  now  the  methods  which  can  be  used 
to  convey  information  from  one  place  to  another.  The 
simplest  way  is  to  send  a messenger  from  one  of  the 
places  to  the  other ; but  the  information  may  also  be 
conveyed  by  handing  it  on  from  person  to  person  placed 
in  suitable  positions.  In  the  first  case  some  material 
substance  must  be  transferred  from  the  one  place  to  the 
other,  but  in  the  second  it  need  not.  These  two 
methods  illustrate  the  two  chief  theories  that  have  been 
proposed  to  explain  light — the  emission  theory  and  the 
wave  theory. 

3.  In  the  emission  theory  proposed  by  Sir  Isaac 
Newton  it  is  supposed  that  sources  of  light  shoot  out 
very  small  particles  in  all  directions  with  great  velocity. 
These  would  rebound  on  striking  any  body,  and  reflec- 
tion would  thus  be  produced,  and  in  passing  from  one 
medium  to  another  they  would  be  deflected,  and  refrac- 
tion would  result.  As  far  as  reflection  and  refraction 
are  concerned,  the  theory  was  fairly  satisfactory ; but 
when  attempts  were  made  to  explain  other  phenomena, 
those  of  polarization,^  for  instance,  very  serious  difficul- 
ties were  encountered,  and  to  surmount  these  many 
additional  hypotheses  were  required.  After  a time  the 
theory  became  so  unwieldy  that,  to  say  the  least,  it 
^ For  these  see  any  treatise  on  optics. 


ON  LIGHT 


5 


must  have  required  a wide  stretch  of  imagination  to 
regard  it  as  representing  the  real  state  of  affairs. 
Besides  this,  there  was  the  difficulty  of  accounting  for 
the  fact  that  a dead-black  surface  which  absorbs  nearly 
all  the  light  falling  on  it  receives  no  increase  of  weight 
on  that  account.  And  lastly,  the  emission  theory  has 
failed  to  stand  the  test  of  a crucial  experiment  devised 
by  Arago.^ 

4.  Newton,  however,  before  he  proposed  the 
emission  theory,  had  entertained  the  idea  of  the  wave 
theory,  but  had  been  unable  to  reconcile  it  with  the 
rectilinear  propagation  of  light.^  For  while  sound, 
which  is  due  to  waves  of  condensation  and  rarefaction 
in  air,  is  able,  to  a great  extent,  to  bend  round  obstacles, 
so  that  they  offer  little  impediment  to  it,  on  the  other 
hand,  light  had,  so  far  as  Newton  could  observe,  no 
similar  power  of  bending,  anything  between  the  object 
and  the  eye  effectually  concealing  the  object.  But  the 
difficulty,  being  to  some  extent  based  on  a false  analogy, 
is  not  so  great  as  it  at  first  sight  appears  to  be ; for  a 
sound  is  not  altered  in  character  by  any  reflection  it 
may  undergo,  e.  g.  at  the  walls  of  a room,  and  thus 
after  such  a reflection  we  recognize  it  as  the  original 
sound,  though  diminished  in  intensity.  But  if  light 
enter  a room  through  a small  opening,  and  is  reflected 
and  scattered  by  the  walls,  we  cannot  thereby  see  the 
object  outside,  though  light  may  penetrate  to  all  parts 
of  the  room. 

At  the  beginning  of  the  century  this  difficulty  about 
the  rectilinear  propagation  of  light  was  finally  cleared 
up  by  the  independent  labours  of  Dr.  Thomas  Young  in 
England  and  of  Augustin  Fresnel  in  France.  This 
great  obstruction  being  then  removed,  the  wave  theory 
made  rapid  progress,  and  in  a few  years  was  accepted 
by  the  majority  of  scientific  men. 

^ Glazebrook’s  Physical  Optics^  p.  121. 

^ Glazebrook,  Presidential  Address  to  Section  A,  British  As- 
sociation, 1893,  reported  in  Nature ^ September  14,  1893. 


6 


PHOTOOxEAPHIC  OPTICS 


The  theory  has  stood  many  very  severe  tests,  both  of 
exjDeriment  and  calculation.  Fresnel  applied  it  to 
explain  the  phenomena  of  diffraction  and  of  double 
refraction,  verifying  most  of  his  results  by  careful 
measurement ; since  his  time  it  has  constantly  been  the 
subject  of  investigation,  so  that  at  the  present  day, 
though  there  still  remain  points  (such  as  astronomical 
aberration)  to  be  cleared  up,  there  is  very  little  doubt 
of  its  truth.  Recently  it  has  received  still  further 
confirmation  from  the  discovery,  by  Clerk-Maxwell  and 
Hertz,  of  the  intimate  connection  between  optical  and 
electrical  phenomena. 

5.  The  second  method  of  conveying  information  is 
that  of  handing  it  on  from  place  to  place  without  the 
transference  of  any  material  substance ; we  have  now 
to  see  how  wave  motion  satisfies  this  condition  and  fits 
in  with  the  other  points  enumerated. 

To  those  who  live  by  the  seaside  there  will  be  little 
difficulty  in  understanding  how  waves  can  travel  for 
considerable  distances  and  preserve  their  identity,  and 
most  people  are  familiar  with  the  effect  of  throwing  a 
stone  into  a still  pond,  thus  causing  waves  to  travel 
outwards  to  great  distances  from  the  disturbance. 
That  a wave  travels  does  not  mean  that  water  is  carried 
with  it,  but  merely  that  an  up-and-down  motion  is 
handed  on  from  particle  to  particle ; if  this  were  not  so 
there  would  be  a tendency  for  water  to  accumulate  on 
any  shore  on  which  waves  were  breaking.  It  is  shown 
in  special  treatises  that  the  rectilineal  propagation  of 
light,  its  reflection,  refraction,  polarization,  etc.,  can  all 
be  accounted  for  in  a natural  and  simple  manner  from 
the  properties  of  wave  motion.^ 

Reflection  of  waves  is  not  at  all  difficult  to  observe. 
For  example,  at  any  long  sea-wall,  such  as  that  over 
which  the  Great  Western  Railway  runs  between 
Dawlish  and  Teignmouth,  there,  when  the  sea  is  not 
too  rough,  the  incoming  waves  and  the  outgoing 
^ Glazebrook’s  Physical  Optics,  chap.  iii. 


ON  LIGHT 


7 


reflected  waves  can  be  clearly  seen  passing  over  each 
other  without  stopping  each  other’s  motion.  Also,  in 
addition  to  the  long  waves,  other  small  waves  and 
ripples,  crossing  and  recrossing  in  all  directions,  will  be 
seen,  illustrating  how  different  beams  of  light  can  pass 
quite  independently  across  the  same  space.  A short 
time  devoted  to  the  study  of  waves  under  these  con- 
ditions will  convey  clearer  and  more  tangible  ideas  of 
wave  motion  than  any  description. 

We  must  now  see  to  what  properties  of  wave  motion 
the  intensity  and  colour  of  light  correspond  ; it  seems 
reasonable  to  expect  that  the  former  should  depend  on 
the  height  to  which  the  wave  rises,  or,  in  other  words, 
on  the  amount  of  disturbance  to  which  the  particles  of 
water  are  subjected.  To  state  this  more  exactly,  call 
the  length  of  the  excursion  of  the  particles  the  ampli- 
tude of  their  motion,  then  the  intensity  is  proportional 
to  the  square  of  the  amplitude  ; this  has  been  verified 
by  the  agreement  of  calculation  with  observation. 

Colour  depends  on  the  frequency  of  vibration  of  the 
wave,  by  which  we  mean  the  time  taken  by  a particle 
to  oscillate  up  and  down  and  then  return  to  its  original 
position  ; violet  light,  which  is  the  most  photograph- 
ically active,  is  composed  of  the  waves  of  shortest 
frequency,  while  red  is  composed  of  slower  waves.  As 
long  as  we  keep  to  one  medium  the  frequency  of  the 
wave  is  proportional  to  its  length,  or  to  the  distance 
between  crest  and  crest.  In  what  follows  we  shall 
restrict  ourselves  to  light  waves  in  air  unless  it  is  other- 
wise stated. 

We  can  therefore  substitute  wave-length  for  fre- 
quency, the  violet  light  having  a shorter  wave-length 
than  red  light.  Experiment  has  shown  that  there  are 
light  waves  both  too  short  and  also  too  long  to  affect 
the  eye,  that,  in  fact,  the  eye  is  sensitive  only  to  a com- 
paratively small  part  of  the  waves  which  come  from  the 
sun. 

Light  of  shorter  wave-length  than  the  violet,  though 


8 


PHOTOGRAPHIC  OPTICS 


invisible,  yet  exerts  a chemical  action,  while  rays  of 
greater  length  than  the  red  are  perceived  as  heat.  Thus 
the  radiation  we  receive  from  the  sun  may  roughly  be 
divided  into  three  parts — the  chemical,  the  luminous, 
and  the  calorific ; but  there  is  no  sharp  line  of  division 
between  these  parts,  they  overlap  each  other,  the  waves 
of  each  being  of  precisely  the  same  nature,  differing  only 
in  length. 

6.  Some  figures  in  connection  with  the  wave  theor}^ 
may  serve  to  render  ideas  more  concrete.  As  stated  above, 
the  velocity  of  light  in  space  has  been  found  to  be  about 
180,000  miles,  or  300,000,000  metres,  per  second,  a 
velocity  which  would  enable  it  to  travel  in  one  second 
three  times  round  the  earth. 

The  unit  by  which  wave-lengths  are  usually  measured 
is  the  tenth  metre,  which  is  10“^®,  or  one  ten  thousand 
millionth  part  of  a metre ; the  wave-length  of  the 
shortest  visible  violet  ray  is  about  3,900  tenth  metres, 
that  of  the  yellow  light  obtained  by  putting  common 
salt  in  the  flame  of  a spirit  lamp  is  5,890  tenth  metres, 
and  that  of  the  red  rays  about  7,640  tenth  metres.  The 
length  of  the  largest  visible  wave  is  thus  barely  twice 
that  of  the  shortest. 

Knowing  the  length  of  a wave  in  air  and  its  velocity, 
we  can  find  how  many  waves  enter  the  eye  in  one 
second,  for  they  will  be  all  the  waves  comprised  within 
the  length  of  three  hundred  millions  of  metres.  If  we 
take  the  mean  wave-length  given  above,  we  get  the 
enormous  number  of  sixty  billions  in  round  numbers. 
This  seems  incredible,  but  the  magnitude  of  the  number 
is  no  primd  facie  evidence  against  its  accuracy,  for  our 
idea  of  time  is  only  relative. 

Prof.  C.  Y.  Boys  ^ has  recently  photographed  a flying 
rifle  bullet  illuminated  b}^  an  electric  spark  which  he 
found  to  last  only  for  one  or  two  millionths  of  a second. 
It  might  be  thought  that  this  is  too  short  a time  for  the 
light  to  produce  any  effect  on  a photographic  plate,  but 
1 Nature,  vol.  xlviii.  pp.  415,  440,  March  2 and  9,  1893, 


ON  LIGHT 


9 


from  what  has  been  said  above,  something  like  sixty 
millions  of  waves  impinge  on  the  plate  during  each 
millionth  of  a second,  which  gives  the  matter  a very 
different  aspect. 

7.  The  Ether. — If  light  consists  of  wave  motion  there 
must  be  some  medium  through  which  it  is  propagated. 
It  is  very  unlikely  that  air  is  this  medium,  for  light 
passes  freely  through  the  best  vacuum  that  has  been 
produced.  Besides  this,  we  cannot  suppose  that  all 
space  is  filled  with  an  atmosphere  such  as  that  we 
breathe,  but  yet  we  receive  light  from  the  sun  and  stars. 
So  we  are  driven  to  the  conclusion  that  there  must  be 
some  medium  which  is  the  vehicle  of  light,  but  which  we 
cannot  directly  perceive.  This  medium  is  called  the 
luminiferous  ether^  and  the  evidence  for  its  existence, 
though  indirect,  is  very  strong.  Various  suggestions 
have  been  made  as  to  its  nature,  and  though  some  of 
them  account  for  a great  deal,  none  meet  all  the 
observed  phenomena.  One  great  difficulty  is,  that  in 
order  to  transmit  light  at  its  enormous  velocity  of 
180,000  miles  a second,  the  ether  must  possess  extreme 
rigidity  ; but  that  in  spite  of  this  it  offers  an  inappreci- 
able resistance  to  the  motions  of  the  planets. 

8.  Some  difficulty  may  be  felt  in  connecting  theory 
with  experience,  and  in  realizing  that  the  sensation  of 
light  is  really  produced  by  waves  entering  the  eye ; but 
it  should  be  remembered  that  all  interpretations  of  sen- 
sations are  the  results  of  education.  We  learn  by 
experience  that  a certain  sensation  is  the  result  of 
certain  external  circumstances,  and  after  a time  we  are 
so  much  accustomed  to  this  association  that  the  sensa- 
tion becomes  involuntary,  and  we  lose  sight  of  the 
intermediate  steps.  For  instance,  the  picture  formed 
on  the  retina  of  the  eye  is  inverted,  yet  we  feel  no 
inconvenience,  and  do  not  recognize  the  fact. 

9.  Physical  and  Geometrical  Optics. — For  complete 
explanations  of  optical  phenomena  we  must  turn  to  the 
wave  theory,  from  which,  provided  our  mathematical 


10 


PHOTOGRAPHIC  OPTICS 


machinery  is  powerful  enough,  we  can  in  most  cases  get 
what  we  want ; this  has  been  done  in  the  case  of  pheno- 
mena called  diffraction,  observed  under  certain  conditions 
when  light  passes  through  small  holes  and  slits.  But 
this  method  would  often  be  very  tedious,  particularly  in 
the  theory  of  lenses,  and  a different  one,  which  is  accur- 
ate enough  in  most  cases,  can  be  employed. 

The  former  method  is  called  physical  optics,  and  the 
latter  method  geometrical  optics.  In  geometrical  optics 
certain  fundamental  laws  derived  from  and  confirmed 


by  experiment  are  taken  for  granted,  and  are  used  as 
the  starting-point  for  investigation,  no  inquiry  being 
made  as  to  the  nature  of  light ; instead  of  waves,  rays 
of  light  are  considered,  these  being  straight  in  a homo- 
geneous medium,  but  reflected  or  refracted  according 
to  definite  laws  when  meeting  an  obstacle  or  another 
medium. 

10.  Reflection  and  Refraction. — When  light  meets  a 
transparent  obstacle,  part  of  it  is  reflected  and  part 
refracted  in  a definite  nanner,  but,  besides  this,  there  is 
an  irregular  reflection  or  scattering  of  the  light  in  all 


ON  LTCxHT 


11 


directions,  due  to  slight  roughness  of  the  surface ; if  the 
obstacle  is  polished  a comparatively  small  part  is  scat- 
tered, and  the  amount  reflected  depends  on  the  angle  at 
which  the  light  is  incident.  All  bodies,  even  if  they  do 
not  reflect  or  refract  regularly,  scatter  light,  and  it  is 
through  this  that  we  see  them.  W^e  must  now  state  the 
laws  of  regular  reflection  and  refraction.  If  a ray  of 
light  P O meet  the  surface  A B (Fig.  1),  draw  the  plane 
C D to  touch  the  surface  at  O,  and  O N the  normal  to 


the  surface  perpendicular  to  C D ; then  if  O Q be  the 
reflected  ray,  the  lines  O N,  O P,  O Q will  all  be  on  one 
plane,  and  the  angles  P O N,  Q O N will  be  equal. 

The  angle  PON  which  the  incident  ray  makes  with 
the  normal  to  the  surface  at  the  point  of  incidence  is 
called  the  angle  of  incidence^  and  similarly  the  angle 
Q O N is  called  the  angle  of  reflection. 

Thus  we  may  state  the  law  of  reflection  : 

The  incident  and  reflected  rays  are  in  the  same  plane 


12 


PHOTOGRAPHIC  OPTICS 


with  the  normal  to  the  surface  at  the  point  of  incidence, 
and  the  angles  of  incidence  and  reflection  are  equal. 

Next,  in  the  case  of  refraction,  let  A B (Fig.  2)  be  the 
boundary  surface  between  the  two  media,  air  and  glass 
for  example,  and  let  the  ray  P O be  incident  at  O ; 
draw  the  tangent  plane  C D to  the  surface,  and  MON 
the  normal.  Let  O Q be  the  reflected  ray,  and  O B 
the  refracted  ray ; the  angle  B O M is  called  the  angle 
of  refraction.  Denote  the  angles  PON,  BOM  of 


incidence  and  refraction  by  0 and  (p.  Experiment 
shows  that  the  relation  between  the  angles  of  incidence 
and  refraction  is  given  by 

ShTh  0 

= a constant  quantity 

sin  (p 

as  long  as  we  keep  to  the  same  substances  ; and  also 
that  O P,  M O N,  and  O B are  all  in  one  plane. 

Thus  we  may  state  the  law  of  refraction  : 

The  incident  and  refracted  rays  are  in  the  same  plane 
with  the  normal  to  the  surface  at  the  point  of  incidence, 


ON  LIGHT 


13 


and  make  with  it  angles  whose  sines  are  in  a constant 
ratio  as  long  as  the  media  are  unchanged. 

When  the  ray  passes  from  air  or  vacuum  (which,  for 
our  purpose,  are  optically  indistinguishable),  the  con- 
stant ratio  is  denoted  by  p,  and  called  the  refractive 
index  of  the  medium  into  which  the  ray  passes,  so  that 
sin  0 = jjL  sin  cj). 

11. — Refraction  through  two  Media.  — We  must 
now  see  what  the  ratio  becomes  when  a ray  passes  from 
one  homogeneous  medium  to  another,  and  neither  of 
them  is  air  ; a case  which  occurs  with  cemented  lenses. 

Let  A and  B (Fig.  3)  be  plates  of  two  transparent 
media  whose  refractive  indices  are  /x  ^ /x  2,  the  faces  of 
the  plates  being  parallel,  and  let  a ray  of  light  P Q R 
S T traverse  the  two  plates  and  emerge  again  into  the 
air ; it  can  be  shown  experimentally  that  the  emergent 
ray  S T is  parallel  to  the  incident  ray  P Q. 

It  will  be  useful  further  on  to  remember  that  if  a 
ray  of  light  passes  through  a plate  of  any  medium  with 
parallel  faces,  and  out  again  into  the  original  medium, 
it  will  emerge  parallel  to  its  old  direction,  but  will 
be  laterally  displaced  ; this  is  an  experimental  fact,  and 
can  be  shown  to  agree  with  the  law  of  refraction.  Let 
the  angles  which  the  different  portions  of  the  ray  make 
with  the  normals  to  the  surfaces  be  as  marked  in  the 
figure,  the  first  and  last  angles  being  of  course  the  same. 
Then  at  incidence  and  emergence 

sin  0 _ sin  0 

' — A. — ^ 1 — J = M 

sin  9 siny 

At  R between  the  two  media 
sin  cb 

- — r = const  = /X  ^ B say. 
sin  Y 

Then 

_ sin  (p  sin  (p  sin  0 _ (x^ 

A B • f — • /I  * / 

Sin  Y sin  u sin  y p ^ 

Or,  when  light  passes  from  one  medium  to  another,  and 


14 


PHOTOGRAPHIC  OPTICS 


neither  is  air,  the  constant  ratio  of  the  sines  of  the 
angles  of  incidence  and  refraction  can  be  got  by  dividing 
the  refractive  index  of  the  second  medium  by  that  of 
the  first  medium,  and  this  is  often  called  the  refractive 
index  between  the  two  substances.  If  /x  2 :>  /x  or  the 
second  medium  is  more  refracting  than  the  first,  then 
(p  > xp  or  the  refracted  ray  is  nearer  to  the  normal 
than  the  incident  ray,  and  vice  versa, 

12.  Total  Internal  Reflection. — When  light  passes 
from  a rare  to  a dense  medium,  the  refractive  index  be- 
tween the  two  is  greater  than  unity ; for  air  and  flint 
glass  it  is  about  1*6,  thus  : 

sin  d = 1*6  sin  (p. 

We  have  then  6 the  angle  of  incidence  given,  and 
require  to  find  (p,  which  we  must  do  from  the  relation, 
sin  0 = y ^ sin  0 ; this  is  always  possible  whatever 
the  value  of  6,  for  sin  9 cannot  be  greater  than  unity, 
and  sin  0/l’6  is  therefore  always  less  than  unity,  and 
an  angle  cp  can  be  found  corresponding  to  all  the  values 
of  sin  (p. 

But,  on  the  other  hand,  when  light  passes  out  from 
the  dense  medium,  we  know  0,  and  have  to  find  0 from 
the  relation 

sin  d = 1*6  sin  (p. 

It  is  obvious  if  sin  (p  is  greater  than  1/1*6,  which  is 
quite  possible,  that  1*6  sin  cp  is  greater  than  unity,  and 
then  no  value  of  9 can  be  found  to  correspond,  for  the 
sine  of  an  angle  cannot  be  greater  than  unity.  This 
means  that  if  the  angle  of  incidence  is  greater  than  that 
given  bysm  cp  = 1/1*6,  called  the  critical  angle  for  the 
substance,  the  ray  of  light  cannot  emerge,  and  is  found 
to  be  totally  reflected.  The  critical  angle  in  this  case 
is  about  3 S'"  41'.  This  phenomenon  of  the  stoppage  of 
light  does  not  occur  suddenly  when  the  critical  angle  is 
reached,  but  some  of  the  light  is  reflected  at  all  angles 
of  incidence,  the  amount  of  it  increasing  rapidly  as 
the  critical  angle  is  approached. 


ON  LIGHT 


15 


We  see  then  that  lenses  should  be  arranged  so  that 
the  angle  of  incidence  of  the  light  on  all  the  surfaces 
may  be  as  small  as  possible,  to  avoid  loss  of  light  by 
internal  reflection. 

Total  internal  reflection  can  easily  be  observed  by 
hanging  a string  into  water  in  a vessel  with  transparent 
sides,  and  examining  the  portion  in  the  water  from 
below  ; it  will  in  most  cases  be  found  impossible  to  see 
the  portion  above  the  surface,  what  appears  to  be  the 
continuation  of  the  string  being  only  the  reflection  of 
that  in  the  water,  as  an  attempt  to  touch  it  will  show. 

We  have  now  given  as  much  about  the  simple  laws 
of  reflection  and  refraction  as  concerns  us ; a more 
detailed  account  must  be  looked  for  in  such  books  as 
DeschaneLs  or  Ganot’s  Physics. 

13.  Measurement  of  Light.  — We  must  now  con- 
sider the  question  of  the  measurement  of  quantity  of 
light  and  intensity  of  illumination,  concerning  which 
we  must  have  clear  ideas  when  we  come  to  deal  with 
relative  exposures  with  various  stops  and  lenses. 

Light  cannot  be  measured  with  the  same  facility  as 
length  and  weight,  but  it  is  none  the  less  a definite  and 
useful  quantity.  We  cannot  exactly  define  what  we 
mean  by  a quantity  of  light,  but  the  conception  pre- 
sents no  practical  difficulty ; we  can  get  some  idea  of 
the  quantity  of  light  which  has  fallen  on  a sensitive 
plate  from  the  density  of  the  deposit  produced  on 
development,  and  provided  the  plate  was  not  over- 
exposed, the  density  is  roughly  proportional  to  the 
quantity  of  light,  per  unit  area,  that  has  fallen  on  the 
plate  during  its  exposure.  But  we  cannot  use  this  as 
an  absolute  measure  of  a quantity  of  light,  for  we 
cannot  estimate  density  except  by  the  quantity  of  light 
which  the  film  intercepts. 

W e must  first  choose  a unit  source  of  light  which  can 
be  used  as  a standard  with  which  to  compare  other 
sources ; this  is  generally  taken  to  be  the  standard 
candle  of  the  Board  of  Trade,  used  in  testing  the 


16 


PHOTOGRAPHIC  OPTICS 


illuminating  power  of  gas.  In  place  of  the  standard 
candle  other  sources  of  light  have  been  proposed  as 
standards,  such  as  the  Harcourt  Pentane  lamp.  The 
absolute  intensity  of  the  standard  is  not  important,  but 
it  should  be  easily  and  exactly  reproducible. 

We  shall  not  be  very  much  concerned  with  absolute 
values  of  sources  of  light,  though  it  may  at  first  sight 
appear  that  this  would  be  the  case,  for  the  photographic 
effect  of  light  depends  not  only  on  its  intensity,  but 
on  its  colour  composition,  which  is  very  difficult  to 
estimate.  In  most  cases  the  intensity  is  judged  from 
experience,  or  if  an  instrument  is  used  it  is  one  whose 
action  depends  on  the  photographic  power  of  the  light ; 
this  will  be  treated  further  on  when  we  come  to  the 
question  of  exposure. 

In  the  following  paragraphs  on  photometry,  the 
sources  of  light  are  taken  to  be  all  of  the  same  colour; 
the  comparison  of,  sources  of  different  colours  being  a 
very  difficult  matter. 

We  now  need  two  definitions  : 

The  unit  quantity  of  light  is  that  quantity  which 
falls  per  second  on  unit  area  of  the  surface  of  a sphere 
of  unit  radius  at  whose  centre  a unit  source  of  light 
(one  candle)  is  placed ; the  source  of  light  being  small 
compared  with  the  radius  of  the  sphere. 

The  intensity  of  illumination  of  a surface  is  the 
quantity  of  light  which  falls  per  second  on  unit  area 
of  the  surface. 

The  alteration  of  the  intensity  of  illumination  of  a 
surface  due  to  a change  of  its  distance  from  the  source 
of  light  can  be  found  from  the  fact  that  light  travels 
in  straight  lines  (Fig.  4).  Let  the  source  of  light  be  a 
candle  L ; take  a piece  of  cardboard  A and  place  it  at 
unit  distance  from  L,  take  a second  piece  B and  place 
it  at  twice  the  unit  distance  from  L,  making  it  of  such 
a size  that  it  is  just  covered  by  the  shadow  of  A.  B 
must  therefore  be  four  times  the  size  of  A. 

A certain  quantity  of  light  falls  per  second  on  A, 


ON  LIGHT 


17 


and  if  this  screen  be  removed  the  same  quantity  of 
light  will  fall  on  B,  which  is  four  times  the  size ; hence 
the  illumination  of  B will  be  only  one-quarter  that  of 
A.  In  a similar  manner  it  can  be  shown  that  the 
illumination  of  a screen  C at  three  times  the  unit 
distance  from  A will  be  only  one-ninth  that  of  A,  and 
we  could  thus  find  the  illumination  at  any  distance 
from  L in  terms  of  that  of  A.  The  law  connecting 
the  intensities  of  illumination  at  different  distances  is 
usually  given  by  saying  that  the  illumination  varies 
inversely  as  the  square  of  the  distance  from  the  source 
of  light.  It  must,  of  course,  be  understood  that  the 
source  must  be  so  small  that  we  may  without  serious 


error  regard  it  as  practically  a point  compared  with 
the  distances  to  be  measured.  If  the  source  is  not 
small  we  must  then  imagine  it  divided  up  into  several 
small  portions,  the  effects  of  these  found  separately,  and 
then  added  together.  We  can  put  results  of  this  article 
in  a s3^mbolical  form,  which  may  be  easier  to  realize. 

14.  The  unit  quantity  of  light  has  been  defined  to 
be  that  which  falls  per  second  on  unit  area  of  a sphere 
of  unit  radius  with  one  candle  at  the  centre.  If  there 
be  L candles  at  the  centre,  L units  of  light  per  second 
will  fall  on  each  unit  area  of  the  sphere. 

The  surface  of  a sphere  of  radius  R feet  is  ^ 4 tt  R^ 

^ TT  is  used  to  denote  the  ratio  of  the  circumference  of  a circle 
to  its  diameter,  and  is  for  most  purposes  given  accurately  enough 
by  22/7.  So  in  any  subsequent  calculations  tt  will  simply  be  an 
abbreviation  for  22/7, 

C 


18 


PHOTOGRAPHIC  OPTICS 


square  feet,  and  the  surface  of  a sphere  of  unit 
radius  is  4 tt  square  feet;  hence  the  light  falling  on 
the  whole  surface  is  4 tt  L units  per  second.  So  that 
the  whole  quantity  of  light  emitted  by  a source  of 
candle  power  L is  4 tt  L units  per  second,  and  the 
intensity  of  illumination  of  the  surface  of  the  sphere 
is  L.  We  have  taken  the  surfaces  to  be  part  of  spheres, 
but,  if  the  surface  is  small  compared  with  its  distance 
from  the  source,  we  can,  with  enough  accuracy,  replace 
it  by  a plane  surface ; but  it  must  be  remembered  that 
if  the  plane  is  large  compared  with  its  distance  from 
the  object  the  illumination  will  not  be  uniform  all 
over  it. 

Let  us  now  find  the  illumination  at  a point  Q,  distant 
T feet  from  L,  and  let  P be  distant  one  foot  from  L ; 
then  by  the  law  of  the  inverse  squares — 

Illumination  at  Q _ L P^  _ 1 
Illumination  at  P L 

Illumination  at  Q = ^ X illumination  at  P = ~ 

Example. — It  is  found  when  printing  on  bromide 
paper  by  contact  that  the  proper  exposure  is  twenty- 
five  seconds  if  the  printing  frame  be  held  at  a distance 
of  three  feet  from  a twenty  candle-power  gas  flame. 
Pind  the  necessary  exposure  at  a distance  of  three  and 
a half  feet  from  a fifteen  candle-power  gas  flame. 

The  total  quantity  of  light  per  unit  area  which  falls 
on  the  exposed  negative  in  both  cases  must  be  the 
same. 

From  the  relation  given  above,  the  illumination  in 
the  first  case  (L/r^)  is  20/9,  and  hence  the  total 
quantity  of  light  which  falls  in  unit  time  on  unit  area 
of  the  surface  in  twenty-five  seconds  is — 

20 

— X 25  units. 

9 

Now  let  t seconds  be  the  proper  exposure  in  the 


ON  LIGHT 


19 


second  case,  then  the  total  quantity  of  light  incident 
15 

will  be  , X t units. 


Hence 


15  ^ 20 


20  X 25  X 49 


9 X 4 X 15 


46  seconds,  nearl3^ 


14a.  Illumination  of  an  Area  oblique  to  the  Incident 
Light. — In  the  cases  considered  above  it  is  assumed 


that  the  surfaces  are  all  at  right  angles  to  the  incident 
light ; if  they  are  not  so  the  illumination  will  not  be 
the  same  as  that  which  we  have  found. 

Let  light  coming  from  a fairly  distant  source  fall  on 
a surface  at  an  angle  of  incidence  ; let  the  plane  of 
the  paper  (Fig.  4a)  be  the  plane  of  incidence.  Consider 
the  light  inside  a small  circular  cylinder  bounded  by 
incident  rays,  of  which  C D,  E F are  the  section;  draw 
F N perpendicular  to  C D. 

Let  I be  the  illumination  of  the  area  D F,  and  T 
the  illumination  that  an  area  placed  in  the  position 
F N would  receive. 


20 


PHOTOGRAPHIC  OPTICS 


The  cylindel-  being  small,  the  areas  F N and  F D 
are  practically  at  the  same  distance  from  the  source  of 
light,  and  hence  their  illumination  will  be  proportional 
to  the  quantity  of  light  received  per  unit  area. 

Now  the  two  areas  v/ould  clearly  receive  the  same 
total  quantity  of  light,  since  either  would  receive  all 
the  light  inside  the  cylinder,  hence  the  quantity  of  light 
per  unit  area  received  by  each  will  be  inversely  pro- 
portional to  its  area. 

But  the  area  F N is  the  projection  of  the  area  F D 
on  a plane  inclined  to  it  at  an  angle  ; hence 
Area  F N = area  F D X cos  cj), 
and  therefore 


I 

I' 


area  F N 
area  F 1) 


cos  (/)  or  1 = 1'  cos  (jj 


Hence  we  see  that  the  illumination  of  an  inclined  area 
is  got  from  that  of  an  area  perpendicular  to  the  incident 
light  by  multiplying  by  the  cosine  of  the  angle  of 
incidence.  It  should  be  noted  that  the  above  result  is 
very  approximately  true  even  if  the  incident  rays 
C D,  E F are  not  parallel,  for  the  cylinder  can  be  taken 
as  small  as  we  like,  and  the  rays  C H,  E F in  conse- 
quence as  near  as  we  like  without  altering  the  reasoning. 

15.  Photometry. — The  law  of  the  inverse  square 
supplies  the  means  of  estimating  the  relative  intensities 
of  the  different  sources  of  light ; descriptions  of  the 
various  instruments  employed  can  be  found  in  text- 
books on  light.  Although  for  economy  of  space  de- 
scriptions of  photometers  are  not  here  given,  yet  their 
study  is  strongly  recommended,  and  as  the  apparatus 
required  is  in  most  cases  very  simple  and  easily  con- 
structed, some  practical  acquaintance  with  their  working 
should  be  acquired. 

16.  Dispersion. — If  a ray  of  white  light  fall  on  a 
prism  of  glass  or  other  refracting  substance,  its  direc- 
tion, as  we  have  seen,  is  altered,  and  experiment  shows 
that  the  emergent  beam  is  no  longer  white,  but  coloured. 


ON  LIGHT 


21 


Sir  Isaac  Newton  made  the  experiment  by  allowing  a 
beam  of  sunlight  entering  a darkened  room  through  a 
small  hole  in  the  shutter  to  fall  on  a prism  with  its 
edge  horizontal,  and  refracting  angle  turned  downwards; 
he  received  the  refracted  ray  on  a screen  behind,  and 
obtained  a coloured  vertical  band,  red  at  its  lower  end, 
and  passing  through  orange  yellow,  green  to  blue, 
indigo  and  violet  at  the  upper. 

This  shows  that  white  light  is  composed  of  light  of 


various  colours,  and  also  that  different  coloured  lights 
are  differently  refracted.  This  phenomenon  can  be 
shown  to  take  place  not  only  with  a prism,  but  with  a 
single  refracting  surface ; let  a ray  P O of  white  light 
strike  the  surface  A B at  O (Fig.  5),  the  refracted  beam 
will  consist  of  rays  of  various  colours  such  as  O Q,  O R, 
O S in  various  directions. 

Produce  P O to  T,  then  the  angles  T O Q,  TO  R, 
T O S through  which  the  rays  are  bent  from  their 


22 


PHOTOGRAPHIC  OPTICS 


original  direction  P O are  called  their  deviations.  If 
some  particular  ray  such  as  O Q be  taken  as  the 
standard  of  reference,  then  the  angles  II O Q,  SOU 
which  the  directions  of  the  other  refracted  rays  make 
with  it  are  called  their  dispersions.  This  term  is  very 
appropriate, ' as  it  conveys  vividly  the  idea  of  the 
spreading  out  which  the  various  rays  undergo  on  re- 
fraction. The  result  of  this  dispersion  evidently  is 
that  rays  of  different  colours  have  different  refractive 
indices.  As  a general  rule  the  red  rays  are  the  least 
bent,  and  have  therefore  the  refractive  index  nearest 
to  unity,  and  the  violet  rays  are  the  most  bent,  and 
have  the  greatest  index,  or  the  red  rays  are  the  least 
and  the  violet  the  most  refrangible.  The  dispersion  is 
by  no  means  the  same  in  all  substances  or  in  all  kinds 
of  glass,  and  in  some  substances  it  is  very  irregular 
indeed.^ 

Dispersion  is  of  vital  importance  in  connection  with 
photographic  lenses,  and  must,  therefore,  be  carefully 
studied.  The  dispersion  produced  by  one  refraction  is 
comparatively  small,  but  it  can  be  increased  by  making 
the  rays  undergo  two  or  more  refractions  at  suitably 
arranged  surfaces.  The  usual  arrangement  is  the  prism, 
which  consists  of  a piece  of  glass,  bounded  by  two 
plane  surfaces,  inclined  at  a suitable  angle.  If  a prism 
be  held  close  to  the  eye,  and  a candle  be  examined 
through  it,  both  the  deviation  and  dispersion  will  be 
very  evident,  for  in  order  to  see  the  candle  through  the 
prism  it  must  be  looked  for  in  a direction  considerably 
different  from  its  actual  direction,  and  also  its  edges 
will  be  vividly  tinted  with  colour,  one  side  being  violet 
and  the  other  red. 

To  make  this  clear,  let  a ray  of  light  C P (Fig.  6) 
strike  a prism  at  P,  and  let  P Q and  P R be  the  paths 
of  the  violet  and  red  rays  respectively  after  the  first 
refraction,  their  dispersion  will  then  be  the  angle  Q P R. 
After  a second  refraction,  let  Q S and  R T be  their 
^ See  Glazebrook’s  Physical  Optics,  chap.  viii. 


ON  LIGHT 


23 


directions,  which,  owing  to  the  increase  of  deviation, 
will  be  more  inclined  to  each  other  than  before. 

To  an  eye  receiving  the  rays  Q S and  R T,  the  red 
rays  will  show  the  object  C as  if  it  were  in  the  direction 
S Q,  and  the  violet  rays  as  if  in  direction  T R ; there- 
fore, if  either  ray  could  be  stopped  conrpletely,  the 
object  would  appear  to  be  either  all  red  or  all  violet, 
or,  in  other  words,  each  colour  gives  rise  to  a separate 
image  of  the  object  C.  If  C is  of  an  appreciable  size, 
then  the  images  of  the  different  colours  overlap,  and 


the  result  is  that  in  the  overlapping  portion  the  colours 
re-combine  and  give  the  original  colour  of  the  object, 
while  there  are  fringes  of  red  and  violet  on  either  side. 
In  this  description  two  colours  only,  red  and  violet, 
have  been  mentioned  for  the  sake  of  simplicity,  but 
there  will  actually  be  images  of  the  object  C due  to  all 
shades  of  colour  in  the  light.  The  image  thus  produced 
is  a jumble  of  all  the  colours  present,  white  at  the 
middle  and  coloured  at  the  edges  ; hence,  if  a prism  is 
to  be  used  to  study  the  nature  of  light,  some  means 
must  be  found  to  separate  the  images  due  to  the  various 
colours. 


24 


PHOTOGRAPH rC  OPTICS 


17.  The  Spectroscope. — The  instrument  which  is 
used  for  the  separation  of  colours  is  called  the  spec- 
troscope. The  following  are 
its  main  outlines  (Fig.  7): 
A slit  A is  formed  between  two 
straight  edges  of  metal  placed 
parallel  to  the  edge  of  the  prism, 
one  of  the  straight  edges  being 
movable,  enabling  the  breadth 
of  the  slit  to  be  altered  at 
pleasure  ; L M is  a lens  so  placed 
that  the  slit  is  at  its  principal 
focus,  and  thus,  as  will  be  shown 
further  on,  it  will  cause  the  rays 
to  issue  in  a parallel  pencil. 
After  the  refraction  the  rays  of 
the  same  colour  are  all  parallel, 
but  the  pencils  for  different 
colours  are  in  different  directions 
(red  and  violet  are,  shown). 
These  rays  are  now  received  by 
a telescope  converged  to  a focus 
and  viewed  through  an  eye- piece, 
the  telescope  being  mounted  so 
that  it  can  be  rotated  to  take 
in  all  the  parallel  pencils  of 
rays  in  turn. 

What  is  seen  is  an  assemblage 
of  images  of  the  slit,  one  due 
to  each  of  the  colours  present 
in  the  light.  With  one  prism 
the  separation  is  not  very  per- 
fect, but  more  prisms  can  be 
used  if  desired,  the  rays  passing 
through  them  in  succession,  but 
the  description  given  is  enough  for  our  purpose. 

If  the  light  from  a highly  incandescent  solid,  such 
as  platinum  wire  raised  to  a white  heat  by  the  passage 


ON  LIGHT 


25 


of  an  electric  current,  fall  on  the  slit,  and  the  appear- 
ance be  examined  through  the  telescope,  it  will  be 
found  to  consist  of  a continuous  band  of  light,  the 
colour  varying  through  every  shade  from  violet  to  red, 
showing  that  an  incandescent  solid  sends  out  light  of 
every  visible  shade.  This  band  of  colour  is  called  a 
spectrum,  and  we  have  just  seen  that  when  the  light 
comes  from  a heated  solid  it  is  continuous,  but  this  is 
not  generally  the  case.  Instead  of  the  heated  solid, 


let  the  light  from  various  metallic  salts,  placed  in 
the  flame  of  a spirit  lamp  or  Bunsen  gas-burner,  be 
examined ; we  now  get  only  a few  bright  lines — for 
example,  potassium  gives  two  reddish  lines,  and  one  in 
the  violet  which  is  difficult  to  see ; lithium  gives  red 
and  yellowish  lines ; strontium  (the  chief  ingredient  of 
red  fire)  gives  a group  of  red  lines  and  one  blue  line, 
while  sodium  gives  one  yellow  line,  which  can  be 
separated  by  powerful  instruments  into  two  close  lines. 
But  if  sunlight  be  used,  quite  a different  sight  is 


26 


PHOTOGRAPHIC  OPTICS 


presented  (Fig.  8),  a band  of  colour  stretching  from  red 
to  violet,  but  crossed  at  intervals  by  dark  lines,  which 
have  been  found  by  careful  measurement  to  be  coinci- 
dent with  the  bright  lines  due  to  many  metals.  These 
dark  lines  are  called  Fraunhofer  lines,  from  their  dis- 
coverer, and/  ure  of  the  utmost  importance  in  spectro- 
scopy, enabling  the  presence,  in  the  sun  and  other  bodies, 
of  many  of  our  elements  to  be  detected.  These  dark 
lines  do  not  concern  us,  except  that  the  principal  ones 


are  denoted  by  letters,  and  are  used  to  indicate  the 
different  parts  of  spectra. 

18.  Visual  Intensity  of  Different  Parts  of  the  Solar 
Spectrum. — We  must  now  compare  the  effect  of  light 
from  various  parts  of  the  solar  spectrum  on  the  eye 
and  on  sensitive  plates  and  papers.  The  curve  in 
Fig.  9,  given  by  Langley,^  shows  the  distribution  of 
light,  as  judged  by  the  eye  ; along  the  horizontal  line 
is  shown  the  arrangement  of  the  spectrum,  the  intensity 

1 S.  P.  Langley.  On  the  cheapest  form  of  light,  Phil.  Mag.y 
1890,  vol.  XXX.  p.  270. 


ON  LIGHT 


27 


at  any  point  being  given  by  the  height  of  the  vertical 
line  at  that  point,  drawn  to  meet  the  curve. 

In  this  and  in  following  diagrams,  the  relative 
intensity  only  is  shown,  and  no  two  diagrams  can  be 
compared  as  regards  absolute  intensity.  We  thus  see, 
from  the  diagram,  that  the  greatest  visual  intensity  is 
about  the  line  E in  the  green,  and  before  the  line  G, 
and  after  the  line  D,  the  intensities  are  comparatively 
small.  We  shall  see  below  that  the  maximum  of 
photographic  action  does  not  coincide  with  this  maximum 
of  visual  effect. 

19.  Measurement  of  Density.^ — To  estimate  the 
photographic  effect  of  various  parts  of  the  solar  spectrum 
we  must  be  able  to  measure  the  density  of  deposit,  in 
a plate,  caused  by  the  light,  for  when  the  plate  is  not 
over-exposed  we  may  consider  the  density  as  very 
approximately  proportional  to  the  total  quantity  of  the 
actinic  rays  that  have  fallen  on  it. 

The  following  methods  and  results  are  those  of 
Abney  : By  the  density  of  a deposit  we  mean  the  pro- 
portion of  the  incident  light  which  it  stops  ; thus  if 
half  the  light  is  stopped  the  density  would  be  one-half, 
and  perfect  opacity  is  denoted  by  unity.  The  following 
arrangement  will  determine  the  quantity  of  light  trans- 
mitted, and  we  can  then  reckon  the  quantity  stopped, 
and  thus  get  the  density  : A piece  of  ferrotype  plate  or 
blackened  cardboard  (Eig.  10),  about  the  size  of  a half- 
plate, is  taken,  and  a square  aperture  A is  made  in  it ; 
a square  B equal  to  A is  marked  on  the  plate,  but  not 
cut  out ; both  A and  B are  then  covered  with  white 
translucent  paper.  If  now  a candle  or  lamp  be  placed 
in  front  of  A (Fig.  11),  both  A and  B will  be  illuminated, 
but  if  a rod  of  suitable  size  be  placed  vertically  in  front 
of  A at  the  proper  distance  it  will  prevent  any  light 


^ “Photo-Chemical  Investigations,”  by  Hurter  and  Driffield, 
Journal  of  the  Society  of  Chemical  Industry,  May  31,  1890,  No.  5, 
vol.  ix. 


28 


PHOTOGRAPHIC  OPTICS 


from  the  front  falling  on  A.  Now  place  a lamp  behind 
the  screen,  then  A is  illuminated  by  the  light  which 
comes  through  from  behind ; the  distances  of  the  two 
lights  can  be  arranged  so  that  A and  B appear  equally 
bright.  If  now  we  place  close  behind  A a piece  of  the 
negative  to  be  examined  it  will  shut  off  part  of  the 
light ; to  render  A and  B again  of  the  same  brightness 
the  lamp  behind  must  be  moved  up  nearer  to  the  screen. 
If  we  measure  the  distance  of  the  lamp  behind  from 


the  screen  in  both  cases  we  can  at  once  calculate  the 
fraction  of  the  whole  amount  of  light  which  the 
negative  transmits.  For  example,  suppose  that  in  the 
first  case  the  light  was  thirty-six  inches  away  from  the 
screen,  and  that  in  the  second  case  it  was  nine  inches 
away  (one-quarter  of  the  former  distance),  then  the 
illumination  of  the  negative  was  sixteen  times  as  great 
as  that  of  the  bare  paper  in  the  first  case ; hence  the 
negative  transmits  only  one-sixteenth  of  the  incident 
light,  and  its  density  is  fifteen-sixteenths. 


ON  LIGHT 


29 


In  some  cases  the  light  coming  through  the  negative 


is  slightly  tinted,  which  makes  it  difficult  to  compare 
the  brightness  of  A and  B,  but  this  may  be  avoided 


30 


PHOTOGRAPHIC  OPTICS 


either  by  dyeing  the  paper  a light  coffee  colour  or  by 
observing  through  pale  canary-coloured  glass.  To  avoid 
all  extraneous  light,  the  experiment  should  be  conducted 
in  a darkened  room,  or  if  this  cannot  be  done  the 
whole  apparatus  may  be  placed  in  a large  box  painted 
dead  black  inside,  and  the  observations  of  A and  B 
made  through  a small  hole ; in  all  cases  care  should  be 
taken  that  light  is  not  reflected  by  surrounding  objects 
on  to  A or  B.  The  actual  size  of  A and  B does  not 
matter,  but  it  is  important  that  they  should  be  equal, 
as  it  is  found  that  correct  estimates  cannot  be  made 
unless  the  whole  quantities  of  light  coming  from  the 
two  squares  are  equal. 

In  estimating  the  density  of  photographs  of  the  solar 
spectrum  a direct  measurement  at  the  different  points 
cannot  easily  be  made  on  account  of  the  rapid  variation 
from  point  to  point ; an  indirect  estimation  must  there- 
fore be  made.  A scale  of  density  is  prepared  by 
exposing  small  squares  along  one  side  of  the  plate,  for 
equal  times  at  different  distances  from  a lamp ; the 
distances  being  chosen  to  give  a regular  scale,  and  the 
absolute  density  of  one  of  these  squares  is  found  as 
already  described.  The  densities  at  different  parts  of 
the  spectrum  are  then  found  by  comparing  them  with 
the  density  of  the  squares  on  the  scale. 

20.  Photographic  Effect  of  the  Solar  Spectrum.  — 
We  must  now  examine  some  diagrams  given  by  Abney. 
Fig.  12  shows  the  effect  of  the  solar  spectrum  on  silver 
bromide.  As  before,  the  vertical  ordinates  show  the 
intensity  of  the  effect  at  an}^  point,  and  both  in  this 
and  other  diagrams  the  principal  Fraunhofer  lines  are 
marked  to  enable  the  portion  of  the  spectrum  in  question 
to  be  easily  recognized.  We  notice  in  this  figure  that 
the  maximum  effect  is  due  to  the  portion  near  the  line 
G.  On  referring  to  Fig.  9 we  see  that  the  whole  of 
the  part  of  the  spectrum  which  acts  on  the  bromide 
produces  only  a feeble  effect  on  the  eye.  Again,  for  a 
mixture  of  silver  bromide  and  silver  iodide  (Fig.  13) 


ON  LIGHT 


31 


the  maximum  effect  is  midway  between  the  lines  F and 
G.  In  both  these  cases,  then,  which  give  a fair  idea 
of  the  capabilities  of  plates  which  are  not  isochromatic, 
the  portion  of  the  spectrum  which  produces  the  photo- 
graphic effect  is  considerably  different  from  that  which 
most  powerfully  affects  the  eye ; hence  the  resulting 
monochromatic  rendering  of  coloured  objects  is  not  a 
true  one.  Violet  and  blue  are  represented  on  the  print 


/ 

/ 

1 

/. 

/■ 

.s? 

• 

t- 

V 

-• 

T 

• . ; 

. 

y 

1 

• ^ 

a 

i: 

Lt\ 

□_ 

400C 

5000 

Fig. 

12. 

as  light-coloured,  while  green,  such  as  grass,  is  much 
more  prominent,  and  red,  such  as  that  of  a soldier’s 
coat,  comes  out  quite  black.  This  explains  many 
things  familiar  to  the  practical  photographer — for 
instance,  sky  and  light  clouds  both  sending  out  light 
rich  in  violet  rays  are  photographically  very  much 
alike,  though  different  to  the  eye,  and  hence  little 
difference  between  them  is  produced  in  a negative ; or 


32 


PHOTOGRAPHIC  OPTICS 


in  the  evening,  when  the  sun  is  low  down,  and  the 
light  has  to  pass  through  a considerable  portion  of  the 
atmosphere  usually  laden  with  vapour,  the  violet  rays 
are  greatl}^  intercepted,  and  the  light  is,  in  consequence, 
very  red,  so  that  although  to  the  eye  there  is  plenty  of 
light,  yet  to  the  sensitive  plate  it  may  be  nearly  dark. 

21.  Ortho  chromatism. — To  obviate  these  defects  in 
the  rendering  of  colour,  it  is  necessary  to  make  the 


plate  sensitive  to  that  part  of  the  spectrum  which 
affects  the  eye.  This  has  been  done  to  a certain  extent 
by  staining  the  film  with  an  organic  compound,  which 
increases  its  absorptive  power  for  yellow  and  green  rays, 
and  in  some  manner  not  perfectly  understood  acts  as  a 
go-between,  and  enables  the  light  to  affect  the  silver 
salts  ; cyanin  blue  and  erythrosin  are  such  compounds. 
In  Fig.  14  are  two  curves  given  by  Abney.  No.  I.  is 


ox  LIGHT 


S3 


the  intensity  curve  for  an  ordinary  mixture  of  silver 
bromide  and  silver  iodide,  and  No.  II.  is  the  intensity 
curve  for  the  same  mixture  when  dyed  with  erythrosin. 
The  contrast  is  very  marked ; in  No.  I.  the  maximum  is 
about  the  G line  in  the  weakly  luminous  portion,  while 
in  No.  II.  the  maximum  has  shifted  to  midway  between 
the  D and  F lines,  well  within  the  luminous  portion. 


Fig.  14. 


though  a secondary  maximum  still  remains  near  the  G 
line. 

These  examples  are  enough  to  show  the  general 
nature  of  the  phenomena ; for  more  detailed  informa- 
tion the  reader  is  referred  to  Abney’s  original  papers, 
or  to  his  book,^  where  extensive  information  is  given 
about  both  plates  and  papers,  or  to  Vogel’s  Das  Licht 
im  Dienste  dev  Photographie  (Berlin,  1894). 

^ Photogmpliij , Text-books  of  Science  Series. 


D 


34 


PHOTOGRAPHIC  OPTICS 


22.  Absorption.  — Whenever  light  passes  through 
any  medium  some  part  of  it  is  absorbed,  even  though  a 
thin  layer  of  the  medium  may  appear  transparent.  In 
inferior  lenses  the  glass  is  sometimes  of  a yellowish 
tinge,  and  though  this  may  not  seem  to  the  eye  to  cut 
off  much  light,  yet  it  intercepts  a considerable  quantity 
from  the  violet  end  of  the  spectrum,  and  this  makes  the 
lens  slow  in  action. 

On  the  other  hand,  absorption  by  means  of  yellow 
glass  is  made  of  use  in  orthochromatic  photography  to 
improve  the  monochromatic  rendering  at  the  expense  of 
speed.  We  have  seen  (Fig.  14,  II.)  that  with  a plate 
dyed  with  erythrosin  there  is  a secondary  maximum  of 
effect  near  the  G line,  and  this  if  uncorrected  will 
produce  an  undue  prominence  of  bluish  objects ; to 
obviate  this,  a yellow  glass  is  interposed  either  before 
or  behind  the  lens,  which  cuts  off  all  rays  beyond  the  F 
line.  Thus  the  secondary  maximum  is  excluded,  and 
the  rays  are  limited  to  those  between  the  lines  D and 
E,  which  are  well  within  the  visual  portion  of  the 
spectrum.  It  is  found  that  even  clear  glass  absorbs 
a great  portion  of  the  violet  rays,  and  hence,  if  extreme 
rapidity  is  required,  some  other  substance  should  be 
used.  Prof.  Boys  employed  a quartz  or  pebble  lens 
when  photographing  a flying  rifle  bullet. 

To  allow  the  process  of  development  to  be  observed, 
it  is  necessary  that  the  light  in  the  dark-room  should 
be  such  as  lies  well  outside  the  photographically  active 
portion  of  the  spectrum,  and  yet  within  the  visual 
portion.  The  only  way  to  ensure  that  the  light,  trans- 
mitted by  coloured  glass  or  medium,  is  of  a safe  nature 
is  to  examine  its  spectrum,  and  compare  it  with  the 
foregoing  diagrams.  As  the  result  of  such  an  examina- 
tion, Abney  recommends  a combination  of  stained  red 
and  ruby  glass  for  absolute  safety,  though  stained  red 
glass  by  itself  is  safe  enough  with  ordinary  care.  A 
very  good  medium  is  the  common  orange  paper  which 
is  used  for  packing  purposes;  three  thicknesses  of  it 


ON  LIGHT 


35 


are  safe  when  sunlight  falls  on  the  window.  Canary 
medium  should  be  used  with  caution,  as  it  allows  too 
much  green  light  to  pass. 

When  artificial  light  is  required,  a suitable  lamp  can 
easily  be  found  among  those  in  the  market,  the  chief 
point  then  being  to  find  one  in  which  the  combustion 
is  good,  and  which  does  not  get  too  hot.  If  electric 
light  is  available,  a lamp  made  of  red  glass  for  this 
purpose  can  be  obtained,  thus  avoiding  the  bother  of 
screens. 

The  subject  of  absorption  is  important  in  considering 
the  sensitiveness  of  plates,  for  the  principle  of  energy 
shows  that  a plate  is  acted  on  by  the  light  which  it 
absorbs,  so  that  the  sensitizing  dye  should,  if  possible, 
be  of  the  same  colour  as  the  rays  which  it  is  required 
to  utilize.  To  render  a plate  sensitive  to  all  rays,  it 
would  therefore  be  necessary  to  use  a black  dye  if  one 
could  be  found  to  act,  though  in  this  case  the  negative 
would  be  useless  unless  the  dye  could  afterwards  be 
bleached. 

23.  Photography  in  Colours. — Attempts  have  con- 
stantly been  made  to  produce  a photograph  showing  the 
natural  colours  of  objects ; we  do  not,  of  course,  mean 
the  commercial  coloured  photograph,  which  is  merely 
painted,  but  one  in  which  all  the  colours  are  produced 
by  purely  photographic  processes.  Even  in  the  early 
days  of  photography  some  colour  effects  were  obtained. 
Sir  John  Herschel  exposed  a sensitive  paper  to  the 
solar  spectrum,  and  obtained  coloured  prints,  but  could 
not  fix  them.  Becquerel,  a little  later,  produced 
coloured  photographs  of  the  spectrum  by  using  silver 
chloride  on  a (polished  ?)  silver  plate,  and  this  is  very 
interesting  in  view  of  recent  results.  M.  Yallot,  in  1890, 
obtained  coloured  prints  on  paper  sensitized  by  a special 
process,  but  could  not  fix  them.  Prof.  Yogel,  Mr.  R. 
E.  Ives,  and  others  have  obtained  coloured  photographs 
by  taking  three  or  four  negatives  with  light  passing 
through  coloured  screens,  and  super-imposing  the  prints 


36 


PHOTOGRAPHIC  OPTICS 


from  these  in  their  respective  colours.  ‘‘  The  method 
consists  in  taking  a red,  a yellow,  and  a blue  negative 
of  objects  on  plates  specially  sensitized  for  colours.  The 
three  negatives  are  then  printed  on  to  one  and  the 
sarae  paper  by  means  of  complementary  coloured  rollers 
on  stones.  In  order  to  obtain  the  colours  exactly  com- 
plementary to  those  of  the  negatives,  the  colours  used 
for  printing  were  either  the  coloured  substances  them- 
selves or  some  substance  whose  equivalence  to  these 
had  been  determined  spectroscopically.’’  ^ In  this 
manner  good  results  have  been  obtained,  though  the 
process  is  necessarily  costly. 

One  curious  case  which  was  the  result  of  accident  is 
worthy  of  mention ; at  a meeting  of  the  Manchester 
Philosophical  Society  of  April  8,  1857,  Mr.  Sidebotham 
communicated  the  following  : “In  the  ordinary  collodion 
negatives  on  glass  we  occasionally  meet  with  examples 
of  partial  natural  colouring,  such,  for  instance,  as  a 
green  tint  on  the  foliage.  I have  had  one  in  which  the 
green  and  red  in  a photograph  of  some  scarlet  geraniums 
were  tolerably  bright,  and  I have  here  on  the  table  a 
landscape  with  trees,  and  a red-brick  house  taken  in 
bright  sunshine,  and  you  will  see  the  green  foliage  and 
red  house  are  tolerably  well  marked  in  colour.”^ 
This  photograph  was  fixed,  and  after  thirty-five  years 
the  colours  still  remained  vivid,  but  all  attempts  to 
reproduce  the  effects  failed.  A great  step  has  recently 
been  made  by  Lippman,  who  has  succeeded  in  taking 
coloured  photographs  on  plates  ; his  method  was  rather 
a surprise,  for  it  depends  on  the  physical  properties  of 
light,  whereas  progress  was  looked  for  mainly  on  the 
chemical  side  by  the  discovery  of  suitable  sensitive 
compounds.  A brief  description  of  the  method  and 
theory  may  be  interesting. 

24.  Lippman’ s Coloured  Photographs. — These  are 
produced  by  taking  the  photographs  in  the  ordinary 

^ Nature,  vol.  xlvi.,  p.  263. 

2 Liverpool  and  Manchester  Journal,  April  15,  1857. 


ON  LIGHT 


37 


way  on  plates  backed  with  some  reflector  such  as  a 
clean  mercury  surface  ; wet  plates  are  used,  as  the  grain 


of  dry  plates  is  too  coarse.  The  photographs  are  de- 
veloped in  the  ordinary  way,  and  the  colours  then 


38 


PHOTOGRAPHIC  OPTICS 


appear  ; the  peculiarity  is  that  no  coloured  pigment  is 
produced,  the  colour  being  due  to  the  peculiar  arrange- 
ment of  the  deposit. 

The  outline  of  the  theory  is  as  follows  : When  waves 
are  incident  directly  on  a reflecting  wall,  the  incident 
waves  interfere  with  those  reflected,  producing  what  are 
called  stationary  waves,  in  which  the  vibration  disturb- 
ance, instead  of  advancing,  remains  at  rest.  The  effect 
produced  is  very  much  like  the  vibration  of  the  string 
of  a musical  instrument. 

This  is  diagrammatically  represented  in  Fig.  15:  at  a 
series  of  points  A,  B,  C,  D,  etc.,  called  nodes,  half  a 
wave-length  apart,  there  is  rest,  while  between  them 
there  is  vibration,  most  intense  at  points  midway 
between  them.  The  wave-length  of  light  being  very 
small,  a great  number  of  these  nodes  occur  in  the  thick- 
ness of  the  film,  and  at  them  no  effect  is  produced  on 
the  sensitive  plate,  but  midway  between  them  chemical 
action  takes  place.  Hence  when  the  plate  is  developed 
the  silver  is  deposited,  not  uniformly  throughout  the 
thickness,  but  in  layers  at  regular  intervals  ; the  dis- 
tances between  the  layers,  being  half  the  wave-length 
of  the  light,  are  different  for  light  of  different  colours. 
When  the  negative  is  viewed  by  reflected  light  the 
photograph  is  seen  in  natural  colours.  There  does  not 
seem  to  be  much  hope  of  making  this  process  commercial, 
as  great  care  is  required  in  the  manipulation,  and  it  is 
obviously  very  difficult  to  produce  anything  like  a 
negative  from  which  prints  can  be  obtained. 


CHAPTER  II 


ELEMENTARY  THEORY  OF  LENSES 

25.  Definition  of  an  Image. — We  have  now  learned 
something  of  tlie  nature  of  light,  and  must  direct  our 
attention  to  the  production  of  pictures.  We  want  to 
be  able  to  throw  a picture  of  the  object  to  be  photo- 
graphed on  a sensitive  plate,  and  we  must  clearly 
understand  what  is  meant  by  this ; a little  consideration 
will  show  that  what  we  have  to  do  is  to  make  the 
illumination  at  each  point  on  the  plate  depend  solely 
on  the  illumination  of  some  one  point  of  the  object, 
and  independent  of  that  at  other  points.  In  fact,  each 
point  of  the  image  must  correspond  to,  and  depend 
solely  on,  some  point  of  the  object,  and  it  is  to  obtain 
this  result  that  lenses  are  employed.  We  can,  however, 
produce  pictures  without  the  aid  of  any  lens  at  all  by 
means  of  a small  aperture,  or  pin-hole,  and  this  we  shall 
consider  first,  as  being  the  simplest  case,  and  giving 
the  opportunity  of  introducing  some  points  of  great 
importance. 

26.  Pin-hole  Photography. — It  is  well  known  that  if 
a hole  be  made  in  the  shutter  of  a darkened  room,  and 
a sheet  of  paper  be  held  near  it,  a picture  of  external 
objects  is  thrown  on  the  paper ; the  phenomenon  is  not 
uncommon,  and  imperfect  images  are  often  cast  when 
the  light  shines  through  cracks  or  slits — for  instance, 
most  people  must  have  noticed,  when  lying  awake  in 
the  morning  with  the  blinds  down,  the  confused  motion 

39 


4o 


PHOTOGRAPHIC  OPTICS 


of  the  masses  of  light  and  shade  on  the  ceiling  caused 
by  anything  passing  outside  the  window. 


The  elementary  explanation  is  not  difficult.  Let  P 
(Fig.  16)  be  a luminous  point,  and  AB  a small  hole  in 


ELEMENTARY  THEORY  OF  LENSES 


41 


a screen  (the  shape  of  the  hole  is  not  of  much  conse- 
quence, but  a circular  one  is  generally  used)  ; the  only 
rays  of  light  admitted  are  those  lying  between  P A 
and  P B,  forming  a small  cone  of  light — this  will 
strike  the  screen  E E behind,  and  produce  a small 
patch  of  light  C D.  If  the  hole  is  small  enough  the 
patch  C D will  not  appear  to  the  eye  to  difier  appreci- 
ably from  a point  of  light,  and  the  illumination  at 
each  point  of  the  screen  will  depend  solely  on  that  of 
the  corresponding  point  of  the  object,  and  a picture 
will  therefore  be  formed. 

27.  Sharpness  of  the  Image. — As  far  as  we  have 
seen  yet,  it  does  not  much  matter  at  what  distance 
behind  the  hole  the  screen  E F is  placed,  as  the  picture 
will  always  be  in  focus ; but  there  is  another  thing  to 
be  considered,  the  separating  or  defining  power  of  the 
arrangement,  on  which  the  sharpness  will  depend. 

When  a picture  smaller  than  the  object  is  produced, 
there  must  naturally  be  some  suppression  of  detail ; if 
the  optical  instrument  were  perfect,  it  would  reproduce 
every  detail  exactly,  so  that  if  the  picture  were  magni- 
fied everything  could  be  seen.  But  this  is  never  the 
case.  We  have  already  seen  that  with  a pin-hole  a 
point  of  light  gives  rise  to  a patch,  not  a point,  of 
light  in  the  picture.  If,  then,  two  points  of  light  in 
the  object  be  near  together,  the  patches  in  the  picture 
produced  by  them  may  be  so  close  as  to  be  mixed  up. 
The  eye  is  no  exception  to  the  general  rule,  as  we 
know  from  experience,  for  we  cannot  distinguish  close 
objects  at  a considerable  distance.  In  practice,  this 
imperfection  of  optical  instruments  is  not  of  any  great 
disadvantage  if  it  can  be  kept  within  limits,  so  that 
the  picture  produced  is,  at  least,  as  good  as  that  shown 
us  by  the  eye.  This  separating  power  of  any  instru- 
ment can  be  studied  by  piercing  two  holes  in  a card- 
board screen,  placing  a lamp  behind,  and  finding  how 
far  off  they  can  be  viewed  through  the  instrument, 
and  yet  appear  distinct  from  each  other. 


42 


PHOTOGRAPHIC  OPTICS 


The  defining  power  of  an  instrument  is  measured  by 


CL  *11. 

Pia.  17. 

the  angle  subtended  at  the  instrument  by  the  distance 


ELEMENTARY  THEORY  OF  LENSES 


43 


between  the  closest  pair  of  objects  that  can  be  separated. 
In  the  case  of  the  eye  objects  must  subtend  an  angle 
of  one  minute,  which  is  about  the  angle  subtended  by 
a length  of  eighteen  inches  set  upright  at  the  distance 
of  a mile. 

Obviously,  if  points  are  to  appear  distinct  in  the 
picture,  the  patches  which  represent  them  must  appear 
to  the  eye  to  be  separated  when  the  picture  is  at  the 
distance  of  distinct  vision,  and  hence  the  smaller  these 
patches  the  better  will  be  the  definition  of  the  instru- 
ment. Wallon  takes  the  diameter  of  the  smallest 
permissible  patch  to  be  ‘01  cm.  or  *004  inch. 

28.  If,  in  the  case  of  a pin-hole,  we  have  two  objects 
P P'  near  together  (Fig.  17),  then  the  corresponding 
patches  CD,  C'  D'  on  the  screen  E F will  overlap,  and 
the  two  images  are  not  separated ; this  will  be  to  some 
extent  remedied  if  the  screen  is  moved  further  back, 
for  then  C D and  C'  D'  will  be  more  separated,  but  at 
the  same  time  the  sizes  of  the  patches  of  light  will  be 
increased,  and  the  advantage  gained  is  doubtful.  When 
the  objects  P P'  are  at  a great  distance  from  the  hole 
the  cones  of  light  become  cylinders,  and  their  sections 
will  be  of  the  same  size  wherever  they  are  cut  by  E F, 
and  hence  in  this  case,  if  we  move  the  screen  further 
back,  the  patches  will  move  further  apart,  but  not 
increase  in  size  (Fig.  18);  so  that  although  according 
to  the  rough  theory  the  position  of  the  screen  does  not 
matter,  yet  we  shall  improve  the  definition  by  putting 
E F at  some  definite  distance. 

The  problem  of  pin-hole  photography  has  been  worked 
out  by  Lord  Rayleigh  from  the  considerations  of 
physical  optics,^  and  he  shows  that  the  patches  of  light 
are  not  sharply  marked,  but  that  the  light  fades  away 
gradually  as  we  proceed  from  the  centre  to  the  edge, 
the  rate  of  fading  depending  on  the  distance  of  the 
screen  E F and  the  diameter  of  the  hole.  The  more 
rapidly  the  light  fades,  the  nearer  together  can  we  bring 
^ PhiL  Mag.^  1891,  vol.  xxxi.  p.  87. 


44 


PHOTOGRAPHIC  OPTICS 


two  patches  without  confusion ; thus  the  definition 


Fig.  18. 

depends  on  the  size  of  the  hole  and  the  distance  of  the 


ELEMENTARY  THEORY  OF  LENSES 


45 


screen.  As  the  result  of  both  theory  and  experiment, 
Lord  Rayleigh  has  given  a relation  between  these 
quantities  for  obtaining  the  best  results. 

If  d be  the  diameter  of  the  hole,  and  f the  distance 
of  the  screen  E F from  it,  the  relation  is 

d^^jf  = X 6*0  inches  = 10“^  X 1’5  cm. 

d and  f being  measured  in  the  first  case  in  inches,  in 
the  second  in  centimetres ; the  light  being  taken  as 
that  coming  from  the  most  photographically  active 
portion  of  the  spectrum. 

M.  Colson  ^ has  written  a pamphlet  on  pin-hole 
photography,  the  results  in  which  do  not  agree  with 
those  given  above,  but  his  analysis  does  not  appear  to 
have  been  as  thorough,  and  besides  this,  his  comparison 
of  a pin-hole  with  a lens,  as  regards  distortion,  is 
faulty. 

29.  Disadvantages  of  Pin-hole  Photography. — Though 
at  first  sight  a pin-hole  may  seem  to  offer  great  advan- 
tages on  the  score  of  economy  and  simplicity,  yet  there 
are  serious  drawbacks.  In  order  to  get  good  definition, 
the  plate  must  be  placed  at  an  inconvenient  distance 
from  the  hole,  and  since  only  a small  quantity  of  light 
is  admitted  the  exposure  must  be  long.  Lord  Rayleigh 
found  that  to  photograph  trees  he  had  to  expose  for  an 
hour  and  a half,  even  in  sunshine.  This  length  of 
exposure  is,  of  course,  quite  prohibitive. 

30.  Role  of  the  Lens. — To  increase  the  quantity  of 
light  a larger  aperture  must  be  used,  and  this  would 
destroy  the  picture,  unless  some  special  apparatus  could 
be  found  to  ensure  that  the  illumination  at  each  point 
of  the  screen  still  corresponds  solely  to  that  at  some 
point  of  the  object ; it  is  for  this  purpose  that  a lens 
is  used.  The  most  important  part  of  our  subject  is, 
therefore,  that  dealing  with  lenses,  and  these,  being 
complicated  and  liable  to  many  defects,  require  very 
careful  study. 


Photogm'pliie  sans  Ohjectif,  Gauthier- Villars,  Paris. 


46 


PHOTOGRAPHIC  OPTICS 


A lens  may  be  defined  to  be  a piece  or  assemblage 
of  pieces  of  glass  bounded  by  surfaces  which  are  portions 


ELEMENTARY  THEORY  OF  LENSES 


47 


of  spheres ; it  has  been  proposed  to  make  the  surfaces 
portions  of  ellipsoids  of  revolution  on  account  of  some 
advantage  which  these  forms  seem  to  offer.  But  it  is 
doubtful,  considering  all  the  corrections  necessary, 
whether  anything  would  be  gained  by  deviating  from 
the  spherical  form,  and  even  if  there  were  any  ad- 
vantage the  labour  of  the  necessary  calculations  and 
the  mechanical  difficulty  of  grinding  would  be  pro- 
hibitive. Astronomical  lens  grinders  finally  correct 
lenses  after  they  are  ground,  testing  the  surface  point 
by  point,  and  rubbing  where  necessary,  and  this  may 
amount  in  some  cases  to  giving  the  surfaces  an  ellip- 
soidal form. 

Lenses  are  made  of  various  types  and  kinds  of  glass, 
and  are  usually  named  according  to  their  shape  (Fig. 
19).  A is  a double  convex  lens,  B double  concave, 
C plano-convex,  D plano-concave,  and  so  on ; the  lens 
E,  bounded  by  two  spherical  surfaces  with  their  con- 
cavities in  the  same  direction  and  thicker  in  the  centre 
than  at  the  edge,  is  called  meniscus. 

For  purposes  of  theory,  lenses  are  divided  into  two 
classes,  thick  and  thin  lenses,  the  thickness  in  the 
latter  being  small  compared  with  their  radii  of  curva- 
ture. These  are  very  rarely  realized  in  practice,  but 
their  consideration  is  easier  than  that  of  thick  lenses, 
and  leads  to  many  useful  results. 

We  shall  therefore  begin  with  thin  lenses,  and  then 
proceed  to  extend  the  theory  to  thick  lenses,  to  which 
it  will  afford  an  introduction. 

We  must  begin  with  the  refraction  of  rays  from  a 
luminous  point,  passing  from  air  into  glass  bounded 
by  a spherical  surface,  and  in  this  chapter  we  shall 
confine  ourselves  to  a small  pencil  of  rays  all  near  the 
line  joining  the  luminous  point  to  the  centre  of  the 
surface.  In  the  next  chapter  the  case  of  larger  pencils 
and  oblique  small  pencils  will  be  treated  ; the  former 
will  be  enough  for  our  immediate  purpose,  and  will 
yield  many  useful  results. 


48 


PHOTOGRAPHIC  OPTICS 


31.  Refraction  at  a Spherical  Surface. — Let  R A 
(Fig.  20)  be  the  section  by  the  plane  of  the  paper  of  the 


ELEMENTARY  THEORY  OF  LENSES 


49 


spherical  bounding  surface,  and  P the  luminous  point. 
To  begin  with,  we  must  make  careful  conventions  about 
the  directions  which  are  to  be  reckoned  positive  and 
negative ; by  doing  this  we  can  avoid  a multiplicity 
of  formulse,  which  leads  only  to  confusion. 

The  following  convention  will  always  be  adhered  to 
in  this  treatise  : 

{a)  In  the  case  of  spherical  surfaces  or  thin  lenses 
all  distances  are  to  be  measured  from  the  point 
in  which  the  axis  cuts  the  surface.  In  the 
case  of  a thick  lens  the  lengths  will  be  measured 
from  two  points  called  nodal  points,  as  explained 
further  on. 

(h)  Lengths  are  reckoned  positive  when  they  are 
measured  from  the  starting-point  in  a direction 
opposite  to  that  of  the  incident  light. 

The  axis  of  a single  spherical  surface  is  that  line 
which  joins  the  centre  of  the  sphere  to  the  middle  point 
of  the  surface ; and  the  axis  of  a lens  is  the  line  joining 
the  centres  of  the  bounding  spherical  surfaces. 

In  all  figures  the  light  is  taken  as  coming  from  the 
right  hand  towards  the  left ; the  positive  direction  is, 
therefore,  from  left  to  right. 

Thus  in  Pig.  20  all  the  lines  AO,  A Q,  A P are  in 
the  positive  direction. 

Let  P,  P,  S be  the  path  of  a ray  of  light  near  the  axis, 
and  let  S R produced  backwards  meet  the  axis  A P 
in  Q ; let  O be  the  centre  of  the  surface. 

Let  AO  = r,  AP  = 'w,  AQ  = ^^,  and  let  p,  be  the 
refractive  index  of  the  glass. 

Let  the  angle  PRO,  the  angle  of  incidence,  be  d, 
and  the  angle  Q R O,  which  is  equal  to  the  angle  of 
refraction,  be  and  let  the  angle  R O A be  a. 

OP  sin  O R P sin  6 sin  6 

Then  = sm  R O P ""  sin  a)  ^ sin  a 

O Q sin  O R Q sin  (f)  sin  (f) 

RQ  sin  ROQ  sin  (180°  — a)  sin  a 

E 


50 


PHOTOGRAPHIC  OPTICS 


^x^=^=m,.-.opxrq=mRPxoq 

RF  OQ,  sin  9 

(Note  that  r is  measured  from  A to  O,  not  vice  versa.) 
Now  OP  = AP  — AO  = ^t  — r,  OQ  = AQ  — AO 
' z=  w — r.  When  the  pencil  of  rays  is  small  and 
confined  near  the  axis^  so  that  the  angle  R P A is  small 
for  all  rays  of  the  pencil^  then  we  have  approximately 

RP  = AP=2^,  RQ  = AQ  = t«; 


and  the  relation  found  above  becomes 

w (u  r)  = fJL  u (w  — r) 
or  [JLur  — wr=(iJL  — l)uw 

or  dividing  hj  u w r we  get — 


fX  1 __  |(X  — 1 

w u r 

This  is  the  relation  connecting  the  distances  from  A 
of  the  points  P,  and  Q the  point  in  which  one  of  the 
rays  cuts  the  axis  after  refraction.  Since  this  relation 
does  not  involve  the  inclination  of  the  ray  to  the  axis, 
it  is  true  for  all  rays  near  the  axis,  and  these  will 
therefore,  after  refraction,  all  pass  through  Q,  which 
may  be  called  the  image  of  P. 

Hence,  if  only  a small  portion  of  the  refracting 
surfaces  be  used,  all  rays,  which  before  refraction  pass 
through  a point  (or  converge  towards  one),  will,  after 
refraction,  either  pass  through  or  converge  towards 
another  point. 

In  our  figure  the  rays,  after  refraction,  will,  if 
produced  backwards,  pass  through  a point,  and  the 
image  is  called  virtual,  the  effect  being  that  to  an  observer 
in  the  glass  the  light  would  appear  to  come  from  Q. 

Some  numerical  examples  will  serve  to  drive  home 
the  convention  of  signs,  and  show  the  meaning  of  the 
relation  obtained. 

Example  /. — Let  the  surface  be  as  on  Fig.  20,  and 
let  r = A O = 6 inches,  = A P = 18  inches,  fx  = 
1*5  required  the  position  of  Q. 


ELEMENTARY  THEORY  OF  LENSES 


51 


We  have 


ii  = 1 4-  ^ ~ ^ 

W u r 


18  ^ 6 


18  ^ 12 


5 

36 


w 


or  w = 


36  X 1-5 
5 


10*8  inches. 


.*.  A Q = w = 10*8  inches,  or  Q is  10*8  inches  from 
the  surface  on  the  positive  side ; that  is,  on  the  same 
side  as  the  object,  and  is  virtual. 

Example  IL — Take  the  same  surface,  but  turned  in 
the  opposite  direction ; the  radius  measured  from  the 
surface  is  now  in  the  negative  direction,  hence  (Fig.  21) 

r = — 6 inches,  = 18  inches,  /a  = 1‘5 
/X  l a--l_l  1_  1 

' ' w u r 18  12  36 


Or 


V 

TE 


= — 36,  .*.  -y  = — 54  inches. 


Or  the  image  is  in  this  case  on  the  negative  side, 
that  opposite  to  the  object,  at  a distance  of  fifty-four 
inches  from  the  surface,  and  it  will  be  real. 


We  have  proved  the  relation 

w It  r 

between  the  distances  of  object  and  image  from  the 
surface ; if  the  luminous  point  from  which  the  light 
comes  is  very  distant  we  may  put 

. li  z=  CO  or  - = 0 and  then 
u 


w fJL  r /X  — - 1 

and  this  value  of  w we  may  conveniently  call  the  “ focal 
length  of  the  surface  ” for  entering  rays,  and  denote  it 
by  the  letter  f. 


52 


PHOTOGRAPHIC  OPTICS 


Fig.  21. 


ELEMENTARY  THEORY  OF  LENSES 


53 


0. 


a 


ar 


Fig.  22. 


54 


PHOTOGRAPHIC  OPTICS 


W T 

Thus  f = j-  and  it  is  proportional  to  r. 

32.  Thin  Lens. — To  get  a thin  lens  put  two  spherical 
surfaces  together,  the  distance  between  them  being 
small  compared  with  their  radii.  Consider  a lens  of 
the  form  in  Fig.  22  ; this  is  not  a usual  shape,  but  it 
is  convenient  for  calculation,  since  its  radii  are  both 
in  the  positive  direction.  Relations  for  other  cases 
can  be  found  by  giving  the  proper  signs  to  all  the 
lengths. 

Let  O and  O'  be  the  centres  of  the  front  and  back 
surfaces  respectively,  and  let  A O = r,  A O'  = s.  Let 
P be  the  object,  Q its  image  by  refraction  at  the  first 
surface,  and  R the  image  of  Q by  refraction  at  the 
second  surface,  so  that  R is  the  image  of  P produced 
by  the  two  refractions. 

Let  AP  = ?^,  AQ  = 'i^,  AR  = 'r,  then  from  the 
last  article 

\i  l_/x— 

W U T f 

But  the  distances  A R and  A Q are  connected 
together  in  the  same  way  as  A P and  A Q,  for  if  R 
be  regarded  as  the  object  Q will  evidently  be  its  image 
by  refraction  at  the  second  surface.  This  is,  in  fact, 
a special  case  of  the  general  principle  that  if  a ray  of 
light  be  reversed,  it  will  exactly  retrace  its  path,  and 
so  object  and  image  are  always  interchangeable  terms. 
The  truth  of  this  statement  will  be  made  evident  by 
examining  the  laws  of  reflection  and  refraction,  by  all 
of  which,  if  a ray  be  exactly  reversed,  it  will  retrace 
its  path.  Hence,  for  R and  Q we  have 

p-  1 /X  — 1 

W V s 

For  convenience  we  may  call the  focal  length 


ELEMENTARY  THEORY  OF  LENSES 


55 


of  the  second  surface  for  refraction  out,  and  denote  it 
by  f\  hence  we  get 

W V s f 

Subtract  the  latter  equation  from  the  former,  and  we 
get 


This  is  the  general  formula  connecting  the  distances  of 
the  object  and  image  from  a thin  lens.  P and  P,  the 
positions  of  object  and  image,  are  called  conjugate  foci. 
It  is  clear  that  there  can  be  any  number  of  such  pairs 
of  points,  for  we  could  take  P anywhere  along  A O O', 
and  then  find  the  position  of  the  image  of  Q from  the 
general  relation. 

33.  Principal  Focus  and  Focal  Length. — Particular 
cases  arise  when  either  object  or  image  is  very  distant ; 

if  the  object  is  distant,  u is  very  large  and  — very 

u 

small,  and  we  can  neglect  it,  hence 


This  shows  that  if  the  object  is  so  distant  that  the 
incident  rays  are  practically  parallel,  they  converge 
after  refraction  to  a point  on  the  axis  at  a distance 
from  the  lens  given  by  the  relation  above ; this  point 
is  called  the  principal  focus  of  the  lens,  and  its  distance 
from  the  lens  is  called  the  focal  length  of  the  lens. 

If  the  focal  length  be  called  F we  have 


and  the  fundamental  relation  becomes 


56 


PHOTOGRAPHIC  OPTICS 


If  the  image  is  distant,  v is  large  and  — small,  hence 


1 


1 r 


= - F 


1 

\r  sj  F’ 

or  the  rays  which,  before  refraction,  converge  to  a 
point  distant  F from  the  lens  on  the  negative  side  are 
parallel  after  refraction.  Both  the  points  so  found  are 
sometimes  called  principal  foci,  but  there  will  be  less 
danger  of  confusion  if  we  restrict  this  name  to  the 
first  point,  calling  the  other  the  second  principal  focus. 
We  now  proceed  to  give  examples  of  the  application  of 
the  formula  found  above  to  lenses  of  various  shapes ; 
in  these  the  two  surfaces  will  be  taken,  always  having 
the  same  radii,  five  and  seven  inches,  the  differences 
being  in  their  arrangement ; /x,  the  refractive  index, 
will  be  taken  as  1‘5. 

{a)  The  form  of  the  lens  is  that  in  Fig.  19  F. 
r = 5 inches,  s = 7 inches,  (jl  = 1*5. 

1 


= (m  - 1) 


/I 

1\ 

. /I 

1\ 

— 

= ‘5  X - — 

\r 

SJ 

\5 

7/ 

1 

F ^ V s/  “ " V5  7/  ^ M 

.'.  F = 35  inches,  or  the  principal  focus  is 
thirty-five  inches  from  the  lens  on  the  positive 
side.  Let  there  be  an  object  sixty  inches  in 
front  of  the  lens,  then  u = 60  and 

1 1.1  1^1  1 


= i + ^ = 


(^) 


60  ‘ 35  22T 

V = 22*1,  showing  that  the  image  is  in  front 
of  the  lens,  and  therefore  vertical. 

Double  concave  lens,  as  in  Fig.  19  B.  Here 
r = b inches,  s = — 7 inches,  [i  = 1*5. 


= (m  - 1) 


1 


= -5x(J  + 


6 

35 


F = 


T 


= 5*83  inches. 


Take  the  object  sixty  inches  in  front  as  before, 


ELEMENTARY  THEORY  OF  LENSES 


57 


*’*  ^ ~ u ~^  35  - 60  35  ” 420 


. -y  = 5*32  in. 


and  the  image  is  again  in  front  and  virtual. 

(c)  Meniscus  as  in  Fig.  19  E.  Here  r = 7 inches, 
s = 5 inches,  jjl  = 1*5. 


F = — 35  inches,  showing  that  the  focal 
length  is  the  same  as  in  example  (a),  but  the 
principal  focus  is  on  the  opposite  side  of  the 
lens.  Take  the  object  sixty  inches  in  front  as 
before. 


1 _ ^ _ __  5 

^ ^ 35  “ 420 


* 


-r  = — 84  in.. 


showing  that  the  image  is  behind  the  lens  on 
the  side  opposite  to  that  of  the  object  and  is 
real. 

(d)  Double  convex  lens  as  in  Fig.  19  A. 

Here  r = — 7 inches,  s = b inches,  = 1*5, 


1\_ 

sj 


— -5  X 


.*.  F = 


35 

T 


— 5*83  inches. 


Or  the  focal  length  is  the  same  as  that  in 
Example  2,  but  negative.  Take  the  object 
sixty  inches  in  front  of  the  lens  as  before,  then 


or  the  image  is  on  the  side  opposite  from  that 
of  the  object  and  real. 

These  four  examples  show  how  the  calculations  are 
made ; if  any  difficulty  is  found  in  understanding  them 
the  reader  is  recommended  to  have  recourse  to  pencil 
and  paper,  and  to  find  out  for  himself  where  the  figures 


58 


PHOTOGRAPHIC  OPTICS 


come  from ; a short  time  thus  spent  will  make  many 
things  further  on  much  easier  to  follow,  for  it  will  be 
impossible  to  give  all  the  calculations  at  full  length. 

Two  points  in  these  examples  should  be  noted  : First, 
that  whenever  the  lens  was  thinner  in  the  middle  than 
at  the  edge  the  focal  length  was  positive,  and  when 
thicker  at  the  centre  than  at  the  edge  negative;  secondly, 
that  when  the  focal  length  is  positive  the  image  is 
virtual,  and  when  negative  real  (the  object  is  supposed 
real  in  both  cases). 

These  remarks  will  be  found  to  hold  good  generally, 
and  it  is  useful  to  bear  them  in  mind. 

34.  Principal  and  Secondary  Axes. — The  line  join- 
ing the  centres  of  the  spherical  bounding  surfaces  of  a 
lens  is  called  its  principal  axis,  and  the  centre  of  the 
lens  is  the  point  in  which  the  principal  axis  meets  it ; 
any  other  line  drawn  through  the  centre  of  the  lens 
is  called  a secondary  axis. 

We  have  hitherto  found  only  the  relation  between 
the  distances  of  conjugate  foci  on  the  principal  axis ; 
we  shall  now  show  that  a similar  relation  holds  good 
for  conjugate  foci  on  a secondary  axis  inclined  at  a 
small  angle  to  the  principal  axis.  The  following  proof 
is  adapted  from  Wallon  : ^ 

Let  P be  a point  near  to  but  not  on  the  principal 
axis ; draw  the  axis  P O and  produce  it  (Fig.  23). 

We  know  that  an  image  of  P is  produced  by  the 
lens,  for  we  could  find  its  image  by  refraction  into  the 
glass  at  the  first  surface,  which  will  be  on  the  line 
joining  P to  the  centre  of  that  surface,  and  we  can 
then  find  the  image  of  this  by  refraction  out  at  the 
second  surface.  Since  all  the  rays  from  P pass,  after 
refraction,  through  one  point,  we  can  find  the  position 
of  its  image,  if  we  can  trace  the  path  of  two  different 
rays,  for  they  must,  after  refraction,  intersect  at  the 
image. 

The  first  ray  to  be  traced  is  P O,  in  the  immediate 

^ L'Ohjectif  Fhotographiqiic, 


ELEMENTARY  THEORY  OF  LENSES 


59 


neighbourhood  of  O ; here  the  two  sides  of  the  lens 
are  practically  parallel  planes,  and  they  are  also  near 


together,  and  the  lens  at  this  point  acts  as  a very  thin 
parallel-sided  plate  of  glass,  which  produces  neither 


60 


PHOTOGEAPHIC  OPTICS 


deviation  nor  sensible  lateral  shift  of  a ray.  Hence, 
all  rays  passing  through  O will  proceed,  after  refraction, 
in  the  same  straight  line  as  before,  so  that  P O will 
pass  undeviated,  and  the  image  of  P must  be  in  P O 
produced,  if  necessary. 

Trace  another  ray  P I inclined  at  a small  angle  to 
the  axis  meeting  the  lens  in  I,  and  let  this,  when  pro- 
duced backwards,  cut  the  principal  axis  in  P'.  Then 
we  may  regard  the  ray  P'  I as  coming  from  P'.  Let 
Q'  be  the  focus  conjugate  to  P'.  P'  I will  clearly,  after 
refraction,  pass  through  Q',  and  the  refracted  ray  will 
be  I Q'. 

If  I Q'  intersect  P O in  Q,  this  point  is  the  image  of 
P ; draw  P M,  Q N perpendicular  to  P"  O Q'.  We  are 
now  going  to  show  that  M and  N are  approximately 
conjugate  foci. 

Draw  a line  through  P parallel  to  the  principal  axis 
to  meet  the  lens  in  H,  and  I Q'  in  K ; since  I is  near 
the  axis,  I O may  be  taken  as  perpendicular  to  P"  Q'  O. 
Since  P H is  parallel  to  P'  Q'  O,  we  get 
P K _ P'  Q' 

~ Vo 

and  by  similar  triangles,  P K Q,  O Q'  Q,  we  get 
OJi'  ^ ^ 

PK  PQ 

Multiplying  these  ratios  together,  we  have,  since  P K 
cancels  out 

O Q'  ^ F Q'  O^Q 

PH  P'  O ^ P Q 

or  transposing 

P Q ^ P'  Q' 

PHxOQ  OQ'xFO 
Now  the  angle  POP'  being  small,  we  may  take 
PQ-:MN,  OQ  = ON,  PH  = OM. 

M N _ P'  Q' 

' * (Tai  X O N “ O Q'  X F O 


ELEMENTARY  THEORY  OF  LENSES 


61 


but  M N = M O - O N,  and  P'  Q'  = P'  0 - O Q' 
M0-0N_P'0-0Q' 

■■  OMxON~OQ'xOP' 

or 

J 1_  1 5 

ON  OM  OQ'  OP'  y-  vsee  S 

where  yis  the  focal  length. 

Hence  M and  N are  conjugate  foci. 

This  is  proved  for  one  particular  kind  of  lens,  but  it 
will  hold  good  for  any  lens  if  we  give  the  various  lines 
their  proper  signs. 

We  therefore  have  the  following  method  of  finding 
the  image  of  a point  P near  the  axis  : Draw  P M per- 
pendicular to  the  axis  and  find  N the  focus  conjugate  to 
M ; join  P O and  produce  it  if  necessary ; at  N erect  a 
perpendicular  to  the  axis,  meeting  the  secondary  axis, 
through  P,  in  Q,  thus  Q is  the  image  of  P. 

35.  Conjugate  and  Principal  Focal  Planes. — Now 
suppose  the  object  to  be  an  extended  one  covering  a 
small  plane  perpendicular  to  the  axis,  let  the  axis  cut 
this  plane  in  P,  and  let  the  image  of  P be  at  Q.  To 
find  the  image  of  the  extended  object  we  must  find  the 
images  of  all  points  on  the  plane,  and  their  assemblage 
will  be  the  image  required ; from  the  last  article  we  see 
that  the  images  of  all  points  in  this  plane  lie  in  a plane 
perpendicular  to  the  axis,  which  is  cut  by  the  axis 
in  Q. 

We  see  then  that  a plane  object  near  the  axis  gives 
rise  to  a plane  image ; the  case  of  a large  object  will 
be  considered  when  we  come  to  the  more  complete 
theory  of  a lens  under  the  head  of  spherical  aberration. 
Such  planes  as  the  above  are  called  conjugate  focal 
planes ; when  the  object  is  very  distant  the  rays  of 
light  coming  from  its  various  points  form  parallel 
pencils,  and  as  long  as  these  have  only  a small  inclina- 
tion to  the  axis  they  will  all  give  rise  to  images  on  a 
plane  through  the  principal  focus  F perpendicular  to 


62 


PHOTOGRAPHIC  OPTICS 


the  axis,  which  is  called  the  principal  focal  plane. 
There  will  of  course  be  a second  principal  focal  plane 


ELEMENTARY  THEORY  OF  LENSES 


63 


corresponding  to  the  second  principal  focus,  such  that 
its  image  will  be  at  a great  distance  from  the  lens. 

36.  Geometrical  Construction  for  the  Image. — In 
§ 34  we  found  the  image  of  P by  tracing  the  paths 
of  two  rays  ; we  can  in  this  way  find  the  position 
of  the  image  of  a plane  object,  for  if  we  can  find  a 
point  on  the  image  we  know  the  plane  on  which 
it  lies. 

The  two  rays  whose  path  is  traced  will,  in  this  case, 
be,  firstly,  the  ray  through  the  centre  of  the  lens  which 
passes  unaltered ; secondly,  a ray  parallel  to  the  axis 
which,  after  refraction,  must  pass  through  the  principal 
focus. 

There  are  two  cases  to  be  considered — (a)  When  the 
focal  length  is  positive ; (b)  when  it  is  negative. 

(a)  Positive  focal  length. — In  this  case  the  principal 
focus  is  on  the  same  side  of  the  lens  as  the 
object ; let  it  be  P (Fig.  24) ; let  the  object  be 
an  arrow  A B which  has  the  advantage  of  hav- 
ing its  ends  dissimilar,  so  that  it  is  evident  at  a 
glance  which  way  up  it  is  drawn.  To  trace 
the  first  ray  join  A O,  for  the  second  draw  A I 
parallel  to  the  axis  to  meet  the  lens  in  I ; this 
ray  must,  after  refraction,  when  produced  back- 
wards, meet  the  axis  in  F ; hence  join  F I 
cutting  A O in  a.  Then  a is  the  image  of  A, 
and  we  can  construct  in  a similar  manner  the 
image  of  6,  and  a h will  be  the  image  of  A B ; it 
is  erect  but  virtual. 

(b)  Negative  focal  length. — In  this  case  F will  be  on 
the  other  side  of  the  lens ; the  construction  will 
be  similar  to  that  of  the  last  case,  and  (Fig.  25) 
the  image  will  be  inverted  but  real. 

We  thus  see  the  reason  of  the  remark  at  the  end  of 
§ 33,  that  if  the  focal  length  of  a lens  is  positive, 
the  image  is  virtual,  but  if  it  is  negative  the  image 
is  real. 

The  former  kind  of  lens  is  called  divergent^  for  it  is 


64 


PHOTOGRAPHIC  OPTICS 


easy  to  see  that  a pencil  of  rays  is  always  spread  out  by 
refraction  through  it,  while  the  second  kind  of  lens 
produces  the  opposite  effect,  and  is  convergent. 


ELEMENTARY  THEORY  OF  LENSES 


65 


37.  Magnification. — In  some  kinds  of  work,  such  as 
copying  or  enlarging,  it  is  necessary  to  know  the  relative 
sizes  of  object  and  image  ; magnification  is  defined  to  be 
the  ratio  of  the  size  of  the  image  to  that  of  the  object 
whether  the  image  be  larger  or  smaller  than  the  object. 
The  case  now  considered  is  that  when  the  object  is 
near ; the  case  when  the  object  is  distant  is  considered 
further  on  (§  56). 

In  the  two  figures  of  the  last  article,  let  A B and 
a h cut  the  principal  axis  in  M and  N respectively ; 
then 


Size  of  image  _ a h 
Size  of  object  A B 


but  — 

V 


u f 


by  similar  triangles, 

O M u '' 

which  gives  — = — 

71  TXj  f 


Size  of  image  _ / 

Size  of  object  '^^  + / 

Example. — An  object  is  placed  three  feet  in  front  of 
a converging  lens  of  6-in.  focal  length  ; here  it  = 36 
inches,  y*  = — 6 inches. 

Size  of  image  _ 6 _ 6 _ 1 

Size  of  object  36  — 6 30  5 

Hence  the  size  of  the  image  is  one-fifth  of  that  of 
the  object ; the  negative  sign  means  that  the  image  is 
inverted,  as  we  know  it  should  be.  The  application  to 
enlargements  will  be  given  later  (§  140).  It  should 
be  carefully  noted  that  the  ratio  above  is  that  of 
the  linear  and  not  areal  dimensions  of  the  image  and 
object ; the  areal  dimensions  will  be  proportional  to  the 
squares  of  the  linear  ; thus  in  the  two  examples  any 
area  in  the  image  will  be  one  twenty-fifth  of  the 
corresponding  area  of  the  object. 

38.  Calculation  of  the  Distance  of  the  Image. — We 
have  found  the  relation  connecting  the  distances  of  the 
object  and  image  from  a lens,  so  that  if  we  know  the 
focal  length  / of  the  lens  and  u the  distance  of  the 

F 


PHOTOGRAPHIC  OPTICS 


object  we  can  find  v the  distance  of  the  image  from 
the  lens. 

In  these  calculations  we  have  to  deal  with  reciprocals/ 
and,  except  with  round  numbers,  the  work  is  apt  to  be 
tedious,  but  it  is  much  simplified  by  the  use  of  a table 
of  reciprocals,  which  reduces  the  work  to  addition  and 
subtraction  only.  Such  tables  are  given  in  many  sets 
of  mathematical  tables,  such  as  Bottomley’s  four  figure 
tables  and  many  engineering  handbooks. 

The  reciprocals  are  usually  calculated  to  four  signi- 
ficant figures,  and  will  give  an  accuracy  of  one  in  a 
thousand,  which  is  all  that  is  generally  required. 

Example. — Find  the  position  of  the  image  formed  by 
a convergent  lens  of  focal  length  5*813  inches  of  an 
object  placed  30*56  inches  in  front. 

Here  / = - 5*813,  u = 30*56. 


1^ 

V 


I 

u 


1 

5*813 


= - *1392  = - 


= *03273  -*1720 

1 

7*183 


V = — 7*183  inches 


or  the  image  is  on  the  opposite  side  from  the  object  at 
a distance  of  7*183  inches  from  the  lens. 

39.  Graphical  Calculations. — Lens  calculations  can 
be  performed  graphically  with  a fair  amount  of  accuracy 
by  means  of  a geometrical  construction.  Let  the 
parallel  straight  lines  A B and  C D (Fig.  26)  be  drawn 
to  meet  B D at  any  angle  (it  will  be  convenient  as  a 
rule  to  make  them  perpendicular  to  B D),  and  at  any 
points  B and  D ; join  A T>  and  C B intersecting  in  P, 
and  draw  P N parallel  to  A B or  C D to  meet  B D in 
N ; then  will 

_!_  = J_  +„L 

P N A B C D 


^ The  reciprocal  of  a number  is  unity  divided  by  that  number 
thus  the  reciprocals  of  2 and  4 are  J and  or  *5  and  *25. 


ELEMENTARY  THEORY  OF  LENSES 


67 


By  similar  triangles  P N D,  A B D we  have — 
DN  PN..,  ,BN  PN 
BD  = BD=^ 


68 


PHOTOGRAPHIC  OPTICS 


By  addition — 

PN  P]Sr_DN  + BN_BD 
aIb  CB  ~ B D B D 


■ ■ A B C D PN 


To  apply  this,  suppose  we  have  to  calculate  1/v 
= l/u  + l/f;  on  any  convenient  scale  make  A B = 

C D = /,  and  construct  as  above,  then  evidently  P N 
will  represent  v. 

If  we  wish  to  calculate  1/v  = 1/u  — l//we  have 
only  to  draw  C D downwards  instead  of  upwards,  and 
construct  as  before  (Fig.  27) ; in  this  case  P N lies  below 


ELEMENTARY  THEORY  OF  LENSES 


69 


the  line  B D,  and  is  therefore  to  be  reckoned  negative, 
showing,  as  we  already  know,  that  when  the  focal  length 
is  negative,  the  image  is  on  the  opposite  side  of  the  lens 
from  the  object  and  is  real. 

Another  advantage  of  the  construction  is  that  it  shows 
at  a glance  the  relative  sizes  of  object  and  image,  for 

Size  of  image  _ _ B N 

Size  of  object  u A B 

In  Fig.  26  the  image  is  erect,  which  is  shown  by  P JN 
being  above  B D ; in  Fig.  27  it  is  inverted.  The  con- 
struction can  be  made*  use  of  for  any  calculations 
involving  the  sum  or  difference  of  reciprocals.  There 
is  a geometrical  property  of  the  figure  which  will  be 
found  very  useful  further  on  when  we  come  to  the 
combination  of  lenses  not  in  contact. 

By  similar  triangles  B P IST,  BCD 

BNT  PN..,  ,DN  PNT 
BD  CD  -^BD  AB 


Combining  these 

BJN  B_D  ^ P_N  A B 
B~D  ^ ITn  " ^D  ^ P^ 


IT  PN 

or  cancelling  = 

^ DN 


A B 


Hence  H divides  B D in  the  ratio  of  A B to  C D. 

40.  Combinations  of  Lenses  in  Contact. — Combina- 
tions of  two  or  more  lenses  are  often  used ; we  must 
therefore  be  able  to  deal  with  them.  We  consider 
only  thin  lenses  in  contact ; the  case  of  thick  lenses 
in  contact,  or  separated  by  a sensible  interval,  will  be 
considered  later  (§  52). 

Let  there  be  two  thin  lenses  in  contact  of  focal 
lengths  /i,  ; let  u be  the  distance  of  an  object  in 

front  of  the  first  lens,  the  distance  of  the  image 
formed  by  this  lens,  and  the  distance  of  the  image 
of  the  first  image  formed  by  this  second  lens,  all 


70 


PHOTOGRAPHIC  OPTICS 


measured  from  the  surfaces  of  the  lenses ; then  as 
before  we  have 


1 

"A 


V 


adding  we  ^et 


« /l  h 


Hence  the  combination  acts  like  a lens  of  focal  length 
F,  where 

F A /2 


for  then 


1 


1 


1 

^2  V F 

The  lens  of  focal  length  F is  called,  the  lens  equivalent 
to  the  two  lenses  in  contact,  and  as  far  as  the  relative 
positions  of  object  and  image  are  concerned  it  could 
replace  them. 

The  calculations  of  F,  when  are  known,  can  be 
made  either  by  means  of  the  table  of  reciprocals  or 
graphically,  the  proper  signs  being  given  to  and 
We  can  extend  the  result  to  any  number  of  thin 
lenses  in  contact ; take,  for  example,  a third  lens,  then 


1 _ 2 - 1 

«3  ®2  ~ A 

Add  this  to  the  last  equation  obtained,  and  we  get 

•i’3  « /l  /2  /s 

showing  that  the  focal  length  F of  the  equivalent  lens 
now  is  given  by 

Ui+i+1 

F A A 

Proceeding  in  this  way,  we  shall  get  the  reciprocal 
of  the  focal  length  of  the  lens  equivalent  to  any 
number  of  thin  lenses  in  contact  by  adding  together 
the  reciprocals  of  their  focal  lengths. 


ELEMENTARY  THEORY  OF  LENSES 


71 


Example. — Find  the  focal  length  of  the  lens  equiva- 
lent to  a combination  of  three  lenses,  two  of  them 
being  converging  and  of  focal  length  six  inches,  and 
the  other  diverging  and  of  focal  length  ten  inches. 


Here /i  = — 6 in.,  f^z=\0  in.,  = — 6 in. 


. I 

“ F 


= - -2334 


- -1667  + -1  - *1667 

_ __  1 

4*284 


F = --  4*284  inches 

which  shows  that  the  combination  is  equivalent  to  a 
converging  lens  of  focal  length  4*284  inches. 

41.  Experimental  Determination  of  Focal  Length. — 
It  is  now  clear  that  the  most  important  thing  to  know 
about  a lens  is  its  focal  length,  and  this  can  be  found 
experimentally  without  knowing  either  the  curvatures 
of  the  faces,  or  the  refractive  index  of  the  glass.  In 
practice,  we  should  first  find  the  focal  length,  then  the 
curvatures  by  the  aid  of  a spherometer,  and  from  these 
deduce  the  refractive  index. 

There  are  many  methods  of  measuring  focal  length, 
which  are  described  in  books  on  optics.^  We  shall 
here  describe  a simple  method  requiring  little  apparatus, 
leaving  the  question  of  a combination  till  we  come  to 
the  subject  of  lens  testing. 

{a)  Lens  of  negative  focal  length. — Fix  the  lens 
upright  in  a suitable  support — an  upright  piece 
of  board  with  a hole  in  it  will  do — and  place 
in  front  of  it  something  to  act  as  an  object — 
a pin  or  a hole  in  a piece  of  metal  with  two 
cross  wires  placed  in  front  of  a lamp  are  suit- 
able— and  behind  it  an  upright  screen  of  card- 
board or  paper.  Move  the  lens  and  screens 
about  till  an  image  of  the  object  is  formed  on 
the  screen,  and  adjust  till  the  image  is  as  sharp 


^ Glazebrook  and  Shaw’s  Practiced  Physics,  edition  4,  § 51,  etc. 


72 


PHOTOGRAPHIC  OPTICS 


as  possible,  then  measure  the  distances  between 
the  lens,  the  image,  and  the  object,  and  from 
these  calculate  the  focal  length.  In  arranging 
the  lens  and  screen,  it  may  be  found  that  an 
image  cannot  be  got.  The  reason  for  this  will 
probably  be  that  the  object  and  screen  are  too 
near  together,  for  it  can  be  shown  that  no 
image  is  possible  unless  the  distance  between 
them  is  at  least  four  times  the  focal  length,  so 
it  is  well  to  start  with  the  object  and  screen 
fairly  far  apart. 

ExamjDle. — It  is  found  that  with  a certain  lens  the 
distances  of  object  and  image  from  the  lens  are  twelve 
inches  and  four  inches  respectively. 

Here  u = v = — A: 

1 __1 

J~  V ~ u~  ~ I 12“  3 

.*./*=  — 3 inches. 

In  this  and  in  all  cases  where  accuracy  is  required 
several  measurements  should  be  made  with  various 
distances  of  object  and  image,  the  focal  length  calcu- 
lated from  each,  and  the  mean  of  the  results  taken. 

(6)  Lens  of  ^positive  focal  length  or  diverging  lens. — 
We  have  seen  that  we  cannot  in  this  case  get 
a real  image  of  a real  object ; we  cannot  there- 
fore proceed  directly  as  above,  but  we  may  get 
over  the  difficulty  by  placing  the  diverging  lens 
in  contact  with  a converging  one  of  known 
focal  length,  so  chosen  as  to  make  the  focal 
length  of  the  combination  negative,  and  there- 
fore equivalent  to  a converging  lens.  The  focal 
length  of  the  combination  is  then  found  as 
before,  and  the  required  focal  length  calculated. 

Example. — A convergent  lens  of  focal  length  six 
inches  is  placed  in  contact  with  a divergent  lens,  and 
the  focal  length  of  the  combination  is  found  to  be 
fifteen  inches. 


ELEMENTARY  THEORY  OF  LENSES 


73 


Here  F = — 15  inches,  /i  = — 6 inches,  y’2  ^ 

The  reader  is  left  to  work  the  question  out,  and  to 
verify  that  ^ = 10  inches. 

42.  Range  of  Focus. — In  outdoor  and  landscape 
work  the  objects  are  often  at  a considerable  distance, 
and  the  image  is  in  consequence  very  near  the  principal 
focus  of  the  lens.  From  the  nature  of  the  formulae 
connecting  the  distances  of  object  and  image  from  the 
lens,  it  is  easy  to  show  that  as  the  object  moves  away 
from  the  lens  the  image  moves  towards  the  lens,  at 
first  rapidly,  but  less  so  as  the  object  gets  to  a con- 
siderable distance,  and  after  that  motion  of  the  object 
produces  very  little  further  displacement  of  the  image. 
Hence  the  images  of  all  objects  beyond  a certain 
distance  lie  fairly  near  together  and  close  to  the 
principal  focal  plane. 

The  following  table  shows  the  relative  distances  of 
object  and  image  for  lenses  of  focal  lengths  of  six 
inches,  four  inches,  and  three  inches,  u being  the 
distance  of  the  object  measured  in  feet,  and  v that  of 
the  image  in  inches  : 


Distance  u of 
object  in  feet. 

Values  of  v corresponding  to  those  of  u for  lenses  of 
focal  length. 

- 6 inches. 

- 4 inches. 

- 3 inches. 

10 

- 6*31 

- 4-14 

- 3-07 

20 

- 6T5 

- 4-07 

- 3*04 

30 

- 6-10 

- 4-04 

- 3-02 

Gt.  distance. 

- 6-00 

- 4-00 

- 3-00 

From  these  examples  it  can  be  seen  that  in  the  case 
of  a 6-in.  lens  the  focussing  screen  would  have  to  be 
moved  three-tenths  of  an  inch  if  the  object  moved  from 
a distance  of  ten  feet  to  a great  distance  away ; for  a 
4-in.  lens  the  movement  would  be  only  T4  inch;  and 


74 


PHOTOGRAPHIC  OPTICS 


for  a 3-in.  lens  only  *07  inch.  As  will  be  seen  further 
on,  such  a small  alteration  in  the  position  of  the 
focussing  screen  as  the  tenth  of  an  inch  will  produce 
very  little  change  in  the  sharpness  of  the  picture. 
Hence,  with  a 6-in.  lens  all  objects  more  than  thirty 
feet  away  will  be  approximately  in  focus  at  the  same 
time.  The  same  will  be  the  case  for  lenses  of  4-in.  and 
3-in.  focus  for  objects  more  than  twenty  and  ten  feet 
distant  respectively.  If,  therefore,  we  use  a lens  of 
fairly  short  focus  for  objects  not  nearer  than  twenty 
feet,  we  can  fix  the  position  of  the  plate,  and  need  not 
trouble  to  focus  in  each  particular  case. 

This  is  the  principle  on  which  hand  cameras  with  a 
fixed  focus  are  made. 

Thick  Lenses. 

43.  Thick  Lenses. — The  theory  of  thin  lenses  is  use- 
ful as  an  introduction,  and  the  calculations  are  for  many 
purposes  accurate  enough  ; and  thin  lenses  in  contact 
have  been  shown  to  present  no  difficulty.  But  for 
accurate  work  we  cannot  neglect  the  thickness,  because 
for  some  lenses  the  relation  connecting  the  distances  of 
object  and  image  is  far  from  accurate,  even  when  the 
lens  is  not  very  thick  ; and  besides  this,  a combination 
of  lenses  separated  by  a sensible  interval  can  in  most 
cases  be  replaced  by  a thick,  but  not  by  a thin,  lens. 

Gauss  has  worked  out  the  theory  of  refraction  through 
any  number  of  media  separated  by  spherical  surfaces 
placed  Avith  their  centres  in  any  positions  along  a 
straight  line.  But  we  have  to  consider  only  a single 
medium  bounded  by  two  surfaces,  and  shall  therefore 
be  able  to  introduce  considerable  simplification.  We 
shall  afterwards  consider  a combination  of  two  thick 
lenses,  and  from  this  the  calculation  can  be  extended 
to  any  number  of  lenses  by  taking  account  of  each  of 
them  in  succession. 

44.  Optical  Centre.  Nodal  Points. — Consider  the 


ELEMENTARY  THEORY  OF  LENSES 


75 


lens  in  Fig.  28.  Let  O,  be  the  centres  of  the  two 
bounding  surfaces,  and  A,  A'  the  points  in  which  tlie 
axis  meets  the  surfaces.  Through  O,  O'  draw  two 
parallel  radii  to  meet  the  corresponding  surfaces  in 
Q and  Q',  join  Q Q',  and  produce  it  to  meet  the  axis 


in  C. 


u 


Fig.  28. 


Then,  by  similar  triangles,  O C Q,  O C'  Q',  we  liave 


This  shows  that  C divides  the  line  joining  O O'  ex- 
ternally in  the  ratio  of  the  radii,  and  this  remains  con- 
stant whatever  the  direction  of  O Q,  O'  Q',  and  hence 
C is  a fixed  point. 


76 


PHOTOGRAPHIC  OPTICS 


Since  O Q,  Q'  are  parallel,  the  two  tangents  to  the 
surfaces  at  Q Q',  which  are  perpendicular  to  the  radii, 
are  parallel ; therefore,  if  Q Q'  be  the  path  of  a ray  in 
the  glass,  the  lens  acts  towards  it  as  if  it  were  a parallel- 
sided plate,  and  the  ray  after  refraction  out  from  the 
glass  will  be  parallel  to  its  direction  before  refraction 
(§  11)  into  it.  This  will  be  the  case  for  all  rays  which 
when  produced  pass  through  C ; hence  all  rays  whose 
directions  in  the  glass  (produced  if  necessary)  pass 
through  C traverse  the  lens  without  undergoing  any 
final  deviation,  though  they  may  be  shifted  parallel  to 
themselves.  The  point  C is  called  the  optical  centre  of 
the  lens. 

Let  r (A  0)  and  s (A'  O')  be  the  radii  of  the  sur- 
faces, e the  thickness  A A',  p the  refractive  index  of 
the  glass. 

Let  us  make  use  of  the  abbreviations  we  have  already 
employed  in  §§  31,  32,  i.e. 


T / T 

p — I p — 1 

/ and  /'  being  what  we  have  called  the  focal  lengths  of 
the  two  surfaces  and  proportional  to  their  radii. 


Then 


CO  _ AO  _ r , CO  _ r 
CO'  A'  O'  s’"  00'  s - r 


.-.  C O = 0 0'=  — ^ (s-r^e)  = r- 

s — r s — r s — r 

Hence  AC  = AO-CO  = 

s - / +y 

and  A'  C = A A'  + A C = — ^ 

s — r / + f 

which  gives  the  position  of  C relative  to  the  two 
surfaces. 

Now  let  N and  N'  be  the  images  of  C,  the  optical 
centre,  due  to  refraction  out,  at  the  two  surfaces  ; that 
is,  let  N be  such  a point  that  rays  diverging  from  it 


ELEMENTARY  THEORY  OF  LENSES 


77 


will,  after  refraction  into  the  glass  at  the  first  surface, 
all  converge  towards  C,  and  N'  the  point  towards  which 
they  will  all  converge  after  refraction  out  at  the  second 
surface. 

Hence  all  rays  which  pass  through  N before  re- 
fraction will,  since  they  pass  through  the  optic  centre, 
emerge  through  N'  after  refraction  parallel  to  their 
original  direction. 

The  two  points  JT  and  N'  are  called  the  nodal  points, 
N being  the  nodal  point  of  incidence  and  N'  the  nodal 
point  of  emergence ; planes  through  N and  N'  perpen- 
dicular to  the  axis  are  called  nodal  planes. 

To  find  the  positions  of  N and  N'  we  have  § 31. 

p _ 1 P — ^ 

aTC  ~ AH  “ r 


for  N and  C are  conjugate  foci  with  respect  to  the  first 
surface.  Hence 


^ _ P _ p — I _ __  f +./ 

AH  AC  “ r ^ ef 

= (§  44) 


AN  = 


- ef 

P'  (/  + /^  + 6) 


/ 


It  may  be  shown  in  a similar  manner  that 

ef 


A'  H'  = 


P f'  + 


Hence  A H : A'  H'  = - /:/'  = - r : 5 
or  A H,  A'  H'  are  numerically  in  the  ratio  of  the 
radii. 

45.  Contrast  of  Thick  and  Thin  Lenses. — In  the 

case  of  the  thin  lens  we  saw  that  all  rays  through  the 
centre  passed  without  deviation,  while  with  a thick 
lens  a ray  through  the  nodal  point  of  incidence  emerges 
undeviated  through  the  nodal  point  of  emergence ; thus 
the  effect  of  the  thickness  of  the  lens  on  such  a ray  is 


PHOTOGRAPHIC  OPTICS 


Fig.  29. 


ELEMENTARY  THEORY  OF  LENSES 


79 


not  to  deviate  it,  but  to  give  it  a lateral  shift.  There 
is  then  some  similarity  between  the  centre  of  a thin 
lens  and  the  nodal  points  of  a thick  lens ; for  in  Fig.  25, 
for  example,  the  line  joining  the  corresponding  points  of 
image  and  object  passes  through  O the  centre  of  the  lens, 
while  (Fig.  29)  we  have  A N a N'  parallel.  We  may 
imagine  that  Fig.  29  is  got  from  Fig.  25  by  cutting  the 
diagram  in  half  by  a line  through  O perpendicular  to 
the  axis,  and  then  sliding  the  two  portions  apart  parallel 
to  each  other  to  a distance  N7  'N\  the  two  points  N 'N' 
now  taking  the  place  of  the  single  point  O.  We  saw 
that  (Fig.  22) 

L ^ 1 = L 

V u ¥ 

where  2/.  = A P,  -y  = A Q,  and  F is  the  focal  length  of 
the  lens,  so  we  should  expect  for  a thick  lens,  if  u is  the 
distance  of  the  object  from  N,  and  v the  distance  of  the 
image  from  'N',  we  should  have  a relation  of  the  form. 

i ~ 1=  -L 

V u ¥' 

That  this  is  actually  the  case  will  be  seen  further  on, 
with  this  difference,  that  the  focal  length  is  not  the 
same  as  for  the  thin  lens. 

46.  Size  of  Image  of  an  Object  placed  in  a Nodal 

Plane. — We  must  first  show  that,  if  an  object  be  placed 
in  a nodal  plane,  its  image,  which  will  of  course  be  in 
the  other  nodal  plane,  is  equal  to  it  in  size. 

Let  P NT,  Fig.  30,  be  an  object  in  the  nodal  plane  of 
incidence ; the  image  of  this,  after  refraction  into  the 
glass,  must  be  in  the  plane  through  the  optic  centre 
perpendicular  to  the  axis  (called  the  principal  plane), 
and  we  can  find  its  size  if  we  can  trace  one  ray ; the 
ray  required  is  P O through  the  centre  of  the  first 
surface,  which  being  incident  normally  passes  unde- 
viated. 

Let  P O meet  the  principal  plane  in  R,  so  that  C R is 
the  image  of  P N by  refraction  at  the  first  surface  ; to 


80 


PHOTOGRAPHIC  OPTICS 


Fig.  30. 


ELEMENTARY  THEORY  OF  LENSES 


81 


get  the  image  of  C R by  refraction  out,  join  R O'  cut- 
ting the  nodal  plane  of  emergence  in  Q,  then  Q N'  will 
be  the  image,  for  the  ray  R O'  passes  out  undeviated. 
We  have  to  prove  that  P N,  Q N'  are  equal.  Since 
the  optic  centre  C divides  the  distance  between  the 
centres  of  the  surfaces  in  the  direct  ratio  of  the  radii 
CO  _.r 
~ s 

_ P N NO  , Q N'  N'  O' 

R C “ C O R C ~ CO' 

PN  PN,,  RC  NO  CO'  NO  CO' 
■ “ iTC  ^ “ C^O  ^ N'  O'  ~ N'  O'  ^ C O 

AO  — AN  s r — AN  s 
""  A'  O'  - AN'  ^ r ""  s - A'  N'  ^ r 

but  since  N A,  N'  A'  are  proportional  to  r and  s (§  44), 
r — A N and  s — A'  N'  are  also  in  the  same  ratio. 

-r  — AN  ^PN  r s 

■ ’■  s - A'N'  y ■ '■  (^N'  ""  s ^ r ^ 
or  P N = Q N' 

Hence  the  object  in  one  nodal  plane  has  an  equal  and 
erect  image  in  the  other  nodal  plane  ; this  means  that 
all  rays  passing  through  P in  one  nodal  plane  will, 
after  refraction,  pass  all  through  Q in  the  other  nodal 
plane,  where  P Q is  parallel  to  the  axis.  From  this 
we  can  find  in  what  point  any  ray  meets  the  nodal 
plane  of  emergence  after  refraction,  when  we  know 
where  it  meets  the  nodal  plane  of  incidence  before 
refraction. 

47.  Construction  for  the  Image. — We  can  now  give 
a construction  for  the  image  analogous  to  that  given 
for  the  thin  lens  (|  36). 

Let  F be  the  principal  focus  of  the  lens,  N and  N' 
the  nodal  points,  AB  the  object  (Fig.  31)  (the  lens 
itself  being  omitted  to  simplify  the  figure) ; we  must 
trace  two  rays  from  A.  First  take  the  ray  A N 

G 


82 


PHOTOGRAPHIC  OPTICS 


Fig.  31. 


ELEMENTARY  THEORY  OF  LENSES 


83 


through  the  nodal  point  of  incidence,  its  direction  on 
emergence  is  parallel  to  A IST ; secondly,  take  the 
ray  A P parallel  to  the  axis,  meeting  the  nodal  plane 
of  incidence  in  P ; by  the  last  section  its  direction  after 
refraction  will  pass  through  Q in  the  nodal  plane  of 
emergence  where  Q N'  = P K,  and  also  it  must  pass 
through  the  principal  focus  F.  Let  the  two  rays  meet 
in  a,  which  is. therefore  the  image  of  A;  similarly  we 
may  find  the  image  of  any  point  of  the  object. 


Let  A B and  a h cut  the  axis  in  U and  V,  and  let 
N U = = v,  N'  F - F 

Then  by  similar  triangles  A N U,  a N ' V 
N U _ ^ _ N'  F 

WY  “ "oV  “ Tv  “ Y F 


u 


Or 


V 


u F — vF  = u V 

Or  - - - = i 
V u F 


Hence  we  have  proved  that  the  distances  of  object 
and  image  from  the  nodal  points  of  a thick  lens  obey 
the  same  law  as  their  distances  from  the  lens  obey  in 
the  case  of  the  thin  lens,  the  focal  length  being  the 
distance  from  the  nodal  point  of  emergence  to  the 
principal  focus. 

To  avoid  any  possibility  of  misunderstanding,  we 
will  state  the  convention  of  signs  as  applied  to  a thick 
lens. 

(а)  The  distance  of  the  object  is  measured  from  the 
nodal  point  of  incidence  to  the  object,  and  that 
of  the  image  from  the  nodal  point  of  emergence 
to  the  image. 

(б)  Lengths  are  reckoned  positive  when  they  are 
measured  from  the  starting-point  in  a direction 
opposite  to  that  of  the  incident  light. 

In  all  figures,  the  light  is  taken  as  coming  from  the 


84 


PHOTOGEAPHIC  OPTICS 


right  towards  the  left ; the  positive  direction  is  there- 
fore from  left  to  right. 

If  we  wish  the  rays  to  emerge  parallel  the  image 
will  go  off  to  an  infinite  distance,  and  'o  — go  orl/'r  = 0, 
and  we  get 

- = -or^^  = — I. 
u F 


Hence,  in  this  case  also  there  is  a second  principal 
focus  F^  on  the  side  of  the  lens  opposite  to  F,  and  at 
the  same  distance  from  N that  F is  from  N'. 

47(x.  We  must  now  consider  a point  which  will  be 
useful  when  we  come  to  deal  with  combinations  of 
lenses  not  in  contact. 

We  can  evolve  the  theory  in  a different  order ; if  we 
have  given  the  two  principal  foci  F and  F'  31) 

with  their  properties,  and  the  fact  that  if  an  object  be 
placed  in  one  focal  plane  it  has  an  equal  image  in  the 
other  focal  plane,  we  can  prove  that  the  lines  A H, 
a N'  joining  the  corresponding  points  of  object  and 
image  to  the  nodal  points  are  parallel,  and  hence  deduce 
the  relations  found  in  the  last  article. 

Let  us  trace  two  rays  from  A,  let  A P parallel  to  the 
axis  meet  the  first  nodal  plane  in  P,  mark  off  Q 
equal  to  P N,  then  this  ray  on  emergence  will  proceed 
in  the  direction  F Q ; also  join  A F'  meeting  the  nodal 
plane  in  R,  after  refraction  it  will  be  parallel  to  the 
axis  as  S R.  Let  the  rays  F Q,  S R meet  in  a,  then  a 
will  be  the  point  of  the  image  corresponding  to  A ; we 
have  to  prove  that  AH,  a H"  are  parallel.  Since  the 
triangles  A N U,  a H'  Y are  both  right-angled,  we 
must  prove  that  the  sides  about  the  right  angles  are 
proportionals,  and  thus  the  triangles  similar,  for  then 
the  angles  A H U,  a N' Y are  equal,  and  thus  AH, 
a H'  will  be  parallel. 


How 


HU 
H' Y 


HU^ 

HF' 


X 


H^F 

H'Y 


forHF'  = H'Fbythe 


properties  of  the  principal  foci. 


ELEMENTARY  THEORY  OF  LENSES 


85 


Also  by  similar  triangles 

NU  AR  PR  N'F  _ QF  _ QN'  _ PN 

~ F'  R “ RN  Wy  ~ Qa  ~ Q S “PR 

]sru_PR  PN_ 

■ N'V  “ RLN  ^ FR  “ RN  “ Vy 


Hence  the  sides  of  the  triangles  A H U,  (x  N'  Y about 
the  equal  angles,  are  proportionals  ; the  triangles  are 
therefore  similar. 

Hence  the  angles  A IST  U,  N'  Y are  equal. 

And  therefore  AH,  a ISl'  are  parallel. 

48.  To  Find  the  Focal  Length. — Let  rays  parallel 
to  the  axis  fall  on  the  lens,  and  let  their  image  by  re- 
fraction at  the  first  surface  be  at  a distance  w from 
the  surface,  then  (§  31) 


M ^ - 1 

tu  r 


f ’ 


w = J 


The  first  image,  therefore,  will  be  at  a distance  w + e 
= f + e from  the  second  surface,  since  e is  the  distance 
between  the  surfaces ; let  v be  the  distance  of  the 
second  image  from  the  second  surface,  then 


1 


V 


p 1 __  /X  --  1 fJL 

w e V s /' 

M I I ^ 

+ e ■*■/'-/+  e +/'  “ f (/+T) 


r (/+^) 

R («+/  + /') 


But  (Fig.  28)  -y  = A'  F 


N'F  = A'F  - A'N'  = 


/'  (/+^) 

M (« +/  +y’0 


ef  ^ ff 

M(e +/+/')  H(e +/+/') 


Hence  if  F is  the  focal  length  of  the  lens 


86 


PHOTOGRAPHIC  OPTICS 


^ ff  11.1.^ 

“ K^+7+7')  ‘’Vf  ■ 7+/’  +77' 

This  is  usually  the  most  convenient  formula  to  work 
with,  but  we  can  express  F in  terms  of  the  radii  and 
thickness  by  putting  in  the  values  of  /,  f\  and  we  get 


(m-  1)^6 

[ITS 


Comparing  this  with  | 33  we  see  that  the  effect  of 
the  thickness  is  to  introduce  an  extra  term  into  the 
expression  for  the  focal  length  depending  on  the  thick- 
ness e ; if  we  make  6 = 0,  or  the  lens  thin,  it  reduces 
to  the  expression  for  the  focal  length  of  a thin  lens. 

49.  Numerical  Examples  for  Thick  Lenses. — We 
shall  take  for  calculation  lenses  similar  to  those 
treated  in  | 33,  but  with  a thickness  of  ‘2  inch,  and 
number  the  cases  to  correspond. 

In  each  case  let  x and  y denote  the  distances  of  the 
nodal  points  of  incidence  and  emergence  respectively 
from  the  corresponding  surfaces,  and  let  the  dimensions 
in  § 33  be  used. 

The  values  of  y and  F (quoted  for  reference)  are 
_ - e/  _ ef 

m(«+/+/0’^  M («+/  + / 0 

, 

A*'  («+/+/  ) 


^ Mr  j.,  ^ -jJ-s 

p,  — 1’  p — 1 

(a)  r = 5 inches,  s = 7 inches,  p = 1*5  inch,  e = 
‘2  inch. 


/ = 
f'  = 


l^T 

p - 1 

— p5 


1-5  X 5 


'5 

1-5  X 7 

•5 


= 15, 


p - 1 

6 + / + /'=  -2  + 15  - 21  = 


= - 21 

5*8 


ELEMENTARY  THEORY  OF  LENSES 


“ 


M +/+/') 

ef' 


•^Xj5 
1-5‘x  5*8  " 
-•2  X 21 


M(e  +/+/')  1-5  X 5-8 

//'  15  X 21 


: *34  inch. 
= ’48  inch. 


F = 


= 36*2  inches. 


Hence  both  nodal  points  are  outside  the  lens  and  in 
front  of  it  (Fig.  32  A). 

(b)  Double  concave  lens. 

r = 5 inches,  s = — 7 inches,  ^ = 1*5  inch,  e = *2 
inch. 


= 15,/  = = 21 

/X  — 1 1 


X = 


e+/+/'  = -2  + 15  + 21  = 
- 6/  -2  X 15 


/tCe  +/+/') 

ef’ 

y = 


1-5  X 36-2 
•2  X 21 


36-2 

= — *055  inch. 


F = 


{e  +/+/')  1*5  X 36*2 

//'  _ 15  X 21 


= *077  inch. 
= 5*80  inches. 


+/  + /')  1-5  X 36*2 

Hence  the  nodal  points  are  both  inside  the  lens,  and 
close  to  the  surfaces. 

(c)  Meniscus  (Fig.  32  C). 

r = 7 inches,  s = 5 inches,  /x  = 1*5  inch,  e = *2  .inch. 


y = 


fJL  r 


= 21,  / = 


— /X  s 


/X—  1 /X—  1 

^+f  + f = *2  + 21  - 15 
-ef  _ -*2  X 21 


- 15 
6*2 


F = 


l^{e+J+f)  • 

= 

^ (^  +y  +/) 

y/ 


1*5  X 6*2 
*2  X 15 


1*5  X 6*2 
15  X 21 


= — *451  inch. 

= — *323  inch. 


+/  + /) 


1*5  X 6*2 


= — 33*87  inches. 


88 


PHOTOGRAPHIC  OPTICS 


Hence  the  nodal  points  are  both  outside  the  lens 
and  behind  it. 

(d)  Double  convex  lens. 

r = — 7 inches,  s = 6 inches,  = l‘b,  e = '2  inch. 


/ 


IX  r 


- 21,  / = ; 


— I Ji  s 


- 15 


fJL—l  fJL—l 

e+/+/  = -2  - 21  - 15- -35*^ 


X = 


y = 

F = 


- ef  _ -2  X 21 

M (e  +/  + /')  ~ 1-5  X 35-8 

ef  _ *2  X 15  _ 

+/  + /')  “ 1*5  X 35-8  “ 

ff  _ 21  X 15 

/X  (e+f  + f)  “.1-5  X 35-8  ■" 


— — *.078  inch. 


*056  inch. 


— 5*85  inches. 


Hence  both  nodal  points  lie  inside  the  lens  and  very 
near  the  surfaces,  and  the  focal  length  is  practically  the 
same  as  that  of  the  corresponding  thin  lens. 

The  lenses  (a)  and  (c)  with  their  nodal  points  are 
shown  on  Fig.  3 2 ; the  nodal  points  of  (6)  and  (d)  are  too 
close  to  the  surface  to  be  shown  on  a figure. 

If  one  surface  of  a lens  be  plane,  it  is  not  hard  to  see 
on  inspection  that  the  optical  centre  lies  at  the  point  in 
which  the  curved  surface  is  cut  by  the  axis  ; and  one  of 
the  nodal  points  being  the  image  of  the  optic  centre 
due  to  refraction  at  the  curved  surface  will  coincide 
with  it. 

50.  Magnification. — Since  the  diagram  for  the  thick 
lens  may  be  got  from  that  for  the  thin  lens  by  dividing 
it  along  the  straight  line  perpendicular  to  the  axis,  pass- 
ing through  the  centre  of  the  lens,  and  then  sliding  the 
two  parts  asunder,  it  is  evident  that  what  has  been  said 
about  magnification  for  a thin  lens  will  hold  good  for  a 
thick  one,  provided  the  distance  (u)  is  measured  from 
the  nodal  point  of  incidence  and  (v)  from  the  nodal  point 
of  emergence. 

Or,  from  Fig.  31, 


ELEMENTARY  THEORY  OF  LENSES 


89 


Size  of  image  a Y N'  TJ  v F 

Size  of  object  A U N U u ?^  + F ^ 

51.  Graphical  Calculation. — Since  the  formulee  for 
the  thick  lens  are  of  the  same  form  as  those  for  a thin 
lens,  we  "can  evidently  make  use  of  the  graphical  con- 
struction already  explained,  if  we  assign  the  proper 
meanings  to  the  various  lengths. 

Note. — It  will  be  useful  further  on  to  have  the 
relation  connecting  the  distances  of  object  and  image 
for  a thick  lens  in  terms  of  the  distances  from  the 
surfaces,  instead  of  the  distances  from  the  nodal  points. 

The  positions  of  the  nodal  points  are  dependent  on 
the  refractive  index,  so  that  they  are  not  the  same  for 
different  colours ; it  is  important  to  bear  this  in  mind 
when  considering  the  chromatic  aberration  of  a thick 
lens. 

Let  u and  v be  the  distances  of  the  object  and  image 
from  the  lens  measured  from  the  front  and  back  sur- 
faces respectively,  r and  6"  the  radii  of  the  front  and  back 
surfaces,  e the  thickness  of  the  lens. 

Then,  with  the  same  convention  of  signs,  the  expres- 
sion required  is 


1 

V 


jx\  r 


This  can  be  deduced  without  much  trouble  from  the 
expressions  given  above. 


Combinations  of  Lenses. 

52.  We  have  already  (§  40)  considered  a combina- 
tion of  two  thin  lenses  in  contact ; we  have  now  to  con- 
sider two  lenses  separated  by  a sensible  interval,  and  two 
thick  lenses  in  contact.  It  will  be  seen  that  these  two 
cases  can  be  dealt  with  at  the  same  time. 

We  shall  take  the  case  of  two  thick  lenses,  then  that 
of  two  thin  lenses  can  easily  be  deduced,  and  in  fact  the 
same  formulae  will  apply,  the  only  difference  being  that 


90 


PHOTOGRAPHIC  OPTICS 


in  the  former  case  we  measure  distances 
from  the  nodal  points,  and  in  the  latter 
from  the  surface  of  the  lens. 

The  complete  treatment  of  the  sub- 
ject involves  a considerable  amount  of 
algebra,  so  we  shall  first  state  the  results 
arrived  at,  and  then  give  a geometrical 
verification  by  the  aid  of  the  graphical 
construction  already  explained. 

53.  Statement  of  Results  of  Combina- 
tion.— Let  there  be  two  thick  lenses  of 
focal  lengths  separated  by  a sens- 

ible interval ; we  shall  show  that  the 
combination  is  equivalent  (both  as 
regards  the  relative  positions  of  object 
and  image,  and  also  as  regards  magnifica- 
tion) to  a certain  thick  lens. 

Let  (Fig.  33)  L^  L2  be  the  nodal 
points  of  the  front  lens  focal  length 
Ml  M2  the  nodal  points  of  the  back  lens 
focal  length  and  let  the  distance 
between  the  lenses  be  given  by  M^  L2  = 6, 
measured  from  M^  to  L2.  Now  let 
N2  be  the  nodal  points  of  the  equivalent 
thick  lens,  and  let  F be  its  focal  length, 
and  let 


Li  Ni  = X,  M2  N2 

- efi 


X = 


^ ^ + /i  + 

^ + /i  + A 

And  if  O be  the  optical  centre  of  the 
equivalent  lens,  then  will 

M,  O = 


y,  then  will 
e/2 


/l  +/2 

54.  To  pass  from  this  case  to  that  of 


Fig.  33. 


ELEMENTARY  THEORY  OF  LENSES 


91 


thin  lenses  we  have  only  to  make  the  pairs  of  points 
Li  L2  and  M2  coincide,  they  will  then  be  the 
positions  of  the  thin  lenses. 

55.  Programme  of  Proof. — To  prove  the  statements 
above,  we  have  to  show  two  things  : 

(a)  That  the  distances  of  the  object  and  image  by 
refraction  through  both  lenses  from  two  fixed 
points  are  connected  by  a relation  of  the  same 
Jorm  as  that  which  connects  the  distances  of 
object  and  image  from  the  nodal  points  with  a 
single  thick  lens. 

(5)  That  the  various  quantities  involved  have  the 
values  stated  above. 

We  have  then,  first  of  all,  to  show  that  the  combin- 
ation has  two  points  resembling  the  nodal  points  of  a 
thick  lens,  which  we  shall  identify  by  the  following 
characteristic  property  of  nodal  points  (§  46)  : 

If  an  object  be  placed  in  one  nodal  plane,  its 
geometrical  image,  by  refraction  through  both  lenses,  lies 
on  the  other  nodal  plane,  and  is  equal  in  size  to  the  object. 

We  have  next  to  show  that  the  combination  has  two 
principal  foci. 

When  these  two  points  are  established,  we  know 
(§  4:7a)  that  any  incident  ray  passing  through  one 
nodal  point  of  the  combination  emerges  through  the 
other  nodal  point,  and  is  parallel  to  its  original  direc- 
tion. The  resemblance  between  the  combination  and  a 
thick  lens  will  then  be  complete. 

55a.  Proof. — In  the  first  place,  the  two  lenses  will 
form  a definite  image  of  any  object,  for  if  we  take  the 
effects  of  the  two  lenses  in  succession,  the  first  lens  will 
form  an  image  of  the  object,  and  the  second  lens  will 
then  form  an  image  of  the  first  image. 

We  see  also  from  this  that  there  must  be  two  points, 
corresponding  to  the  principal  foci  of  a single  lens,  to 
which  rays  which  are  parallel  to  the  axis  before  refrac- 
tion converge  after  refraction,  or  from  which  the  rays 
proceed  which  are  parallel  after  refraction. 


92 


PHOTOGRAPHIC  OPTICS 


We  shall  take  the  case  of  two  diverging  lenses,  that 
the  focal  lengths  may  be  positive  ; any  other  case  can 
be  got  by  giving  the  focal  lengths  their  proper  signs. 

Let  us  now  determine  whereabouts  the  nodal  points 
of  the  combination — if  they  exist — must  lie.  Turning 
to  Fig.  24,  we  see  that  the  image  of  a distant  object, 
formed  by  a concave  lens,  is  smaller  than  the  object, 
and  this  will  always  be  the  case  as  long  as  the  object  is 
real. 

For  if  V be  the  distances  of  the  object  and  image 
from  the  lens  of  focal  length  /,  we  have  as  usual 

111  111 

= - or  - = - + ^ 

V U J V u J 

and  f is  positive,  so  v must  be  less  than 

Also  by  § 50 

Size  of  image  v 
Size  of  object 

hence  the  image  is  less  than  the  object. 

Again,  b}^  refraction  at  the  second  lens  the  size  of  the 
image  will  be  still  further  reduced.  Hence,  if  we  are 
to  have  the  final  image  equal  to  the  object,  the  object 
cannot  be  in  front  of  the  first  lens,  but  must  be  virtual 
and  behind  it ; similarly,  the  equal  image  cannot  be 
behind  the  second  lens. 

We  conclude,  therefore,  that  with  two  concave  lenses 
the  nodal  points  of  the  combination  must  lie  between 
those  of  the  component  lenses,  as  shown  in  Fig.  33. 

We  must  now  turn  to  graphical  methods  to  prove  the 
existence  of  the  nodal  points  Ni,  ^2,  and  to  determine 
their  position. 

To  the  straight  line  B D (Fig.  34)  erect  perpendi- 
culars A B ( = y\),  C D ( = produce  these  to  E and 
F,  so  that  A E = C F = e ; join  A D,  B F intersecting 
in  L/,  and  C B,  D E intersecting  in  M2^,  and  let  B C, 
A D intersect  in  O'. 

Drop  perpendiculars  L/  F^,  M2'  F2,  O'  B to  B D,  and 
let  Jji  P\  intersect  B C in  Ni',  M2"  F2,  intersect  A D 


ELEMENTARY  THEORY  OF  LENSES 


93 


in  'N2  and  O"  R produced,  intersect  B F,  D E in  H and 
K respectively. 

First  find  the  positions  of  the  principal  foci  of  the 


94 


PHOTOGRAPHIC  OPTICS 


combination  ; if  parallel  rays  strike  the  first  lens  they 
will  pass  after  refraction  through  its  principal  focus 
distant  /j,  or  A B from  the  nodal  point  of  emergence. 
This  point  will  be  distant  or  E B from  the  nodal 

point  of  incidence  of  the  second  lens  ; the  image  of  this 
point  by  refraction  through  the  second  lens  of  focal 
length  is,  by  the  graphical  construction,  at  a distance 
M2^  Eg  from  the  nodal  point  of  emergence. 

Again,  by  symmetry  it  can  be  seen  from  the  figure 
that  the  rays  emanating  from  a point  distant 
from  the  second  lens  will  after  refraction  by  both  lenses 
be  parallel. 

It  is  evident  also  that  M2'  F2  must  represent  a length 
measured  in  the  positive  direction  and  L^'  F^  is 
measured  in  the  negative  direction,  for  the  two  princi- 
pal foci  are  always  on  opposite  sides  of  the  nodal  points. 

Next  consider  the  lengths  and  O'  H ; is 

at  the  intersection  of  A O and  B H,  so  that 

_J_  =^  + + ^ 

V Ni'  O'  H A B O'  H yi 

1,1  1 

or  — —7-—  + — , = — 

O'  H ^ L/  N/ 

Hence  if  represents  a length  in  the  negative 

direction,  an  object  at  that  distance  from  the  first  lens 
will  have  an  image  due  to  that  lens  at  a distance  — O'  H 
from  it. 

We  must  find  the  distance  of  this  image  from  the 
nodal  point  of  emergence  of  the  second  lens,  by  parallels. 


0'H_ 

B O' 

_ BR 

= 

• O'H  — 1 

CF 

¥c 

f\  +^2 

A + A 

O'K  _ 

DO' 

_ HR 

/2 

• O'  K-- 

AE 

DA 

HB 

/i  + h 

A +A 

.-.  O'  H + O'  K = e or  O'  K = e - O'  H 

Hence  O'  K is  the  distance  of  the  image  from  the  nodal 
point  of  incidence  of  the  second  lens.  Again  by  con- 


ELEMENTARY  THEORY  OF  LENSES 


95 


struction  is  the  intersection  of  C O'  and  D K,  so 
that 


O'  K C D O'  K 

1 1 1 

or  — = — 

m;  n/  ok 

Hence  the  second  image  found  by  refraction  at  the 
second  lens  is  at  a distance  M2'  Ng'  from  the  nodal  point 
of  emergence  of  that  lens.  If,  then,  there  be  a virtual 
object  at  distance  L/  NT/  behind  the  nodal  point  of 
incidence  of  the  first  lens,  it  will  have  an  image  by  re- 
fraction through  both  lenses  at  a distance  M2'  H2^ 
front  of  the  nodal  point  of  emergence  of  the  second 
lens ; also 

Size  of  object  _ L/  N^' 

Size  of  first  image  O'  H ’ 

Size  of  first  image  _ O'  K 
Size  of  second  image  M2'  ^2^ 

Size  of  object  _ L/  O'  K / x 

Size  of  final  image  O'  H ^ M2'  ^2' 


BN, 


CF 

BC 

m;  n/  _ 

dn; 

AE 

DA 

L/N/  = - 

e/i 

B D + /i  + c/2) 


(§  39) 


DF2'  _ 


. WNil  also  ^ _ 

U O'H  /i+/, 


DB 

, M2'n;  = 

— 


(e  + /2  + /i) 
_ e/2 

2 


e + /i  + /2 

/1+/2  Jx 

e/i  A 


Hence,  since  the  product  of  these  latter  ratios  is 
unity,  we  see  from  (a)  that 

Size  of  image  = size  of  object. 

Thus,  the  points  distant  — L^'  N^'  from  the  nodal 
point  of  incidence  of  the  front  lens,  and  M2'  N2'  from 


96 


PHOTOGEAPHIC  OPTICS 


the  nodal  point  of  emergence  of  the  second  lens,  fulfil 
all  the  conditions  for  being  the  nodal  points  of  the 
combination  ; we  therefore  conclude  that  such  points 
exist,  and  that  the  combination  can  be  replaced,  as  far 
as  the  positions  of  object  and  image,  and  size  of  object 
and  image,  are  concerned  by  a single  thick  lens. 

The  two  figures  (33  and  34)  have  been  lettered  to 
correspond,  hence 


- Li  Ni  = - l;  n/  = 


- ^fi 

6 +yi  +^2 
e/2 


e +/i  +^2 


The  focal  length  of  the  combination  is  the  distance 
of  the  principal  focus  from  N2,  the  nodal  point  of 
emergence,  or  of  the  second  principal  focus  from  the 
nodal  point  of  incidence. 

These  lengths  are  evidently  given  in  the  diagram  by 
Ni'  and  N2'  F2,  and  we  can  show  that  these  are 
equal,  as  they  should  be,  for 

F/  _ CD  _A  . ^ .-p,  V 
Lj  C F e’  ^ ^ e e 


.-.N/  F, 


/1/2 

^ +/l  +/2 


and  this  being  symmetrical  with  respect  to  fi  and 
we  shall  evidently  get  the  same  value  for  N2  F2' 

It  is  evident  since  O'  K and  O'  H give  the  position 
of  the  image  produced  by  the  first  lens  of  an  object 
placed  at  that  these  lengths  give  the  position  of  the 
optical  centre  of  the  combination,  and  hence 

0Nj  = 0'K  = ^^^0N2  = 0'H= 

Jl  2 “T  /2 


This  completes  the  proof  of  all  the  statements  in 
§53. 

56.  Combinations  in  General. — We  have  shown 
how  to  find  the  lens  equivalent  to  two  lenses;  if 


ELEMENTARY  THEORY  OF  LENSES 


97 


there  are  three  or  more  lenses,  these  can  be  taken  in 
pairs,  and  replaced  by  equivalent  thick  lenses,  and 
these  can,  in  turn,  be  taken  in  pairs  and  replaced  by 
other  equivalent  lenses,  till  we  finally  arrive  at  a single 
thick  le'ns  which  is  equivalent  to  the  whole  system. 
Thus  we  see  that  any  combination  of  lenses  can  be 
replaced  by  a single  thick  lens.  This  lens  may  not,  in 
all  cases,  be  such  as  can  be  conveniently  realized  in 
practice,  but  this  is  not  often  required  ; its  use  is  to 
simplify  calculations. 

It  must  be  clearly  understood,  when  we  say  that  a 
single  lens  is  equivalent  to  a combination,  we  mean 
only  that  it  will  act  in  the  same  way  as  the  combination 
as  regards  the  relative  positions  and  sizes  of  object  and 
image  ; but  there  are  other  properties  of  combinations 
important  in  photography,  such  as  corrections  for 
spherical  and  chromatic  aberrations  treated  of  in  the 
next  chapter,  which  cannot  be  reproduced  by  a single 
lens. 

57.  Reversibility  of  Optical  Instruments.  From 
what  we  know  about  the  nodal  points  of  lenses  and 
principal  foci,  it  is  clear  that  if  a lens  be  reversed  so 
that  the  positions  of  the  nodal  points  are  interchanged, 
no  change  in  the  position  of  the  image  of  any  object 
will  be  produced  ; and  as  the  combination  is  equivalent 
to  a single  thick  lens  it  can  be  reversed  in  like  manner. 

This  is  evident  at  first  sight  when  combinations  are 
symmetrical,  but  not  in  other  cases. 

68.  Numerical  Example. — Dallmeyer’s  wide-angle 
landscape  lens. 

This  objective  consists  of  three  lenses  in  contact. 
The  component  lenses  are  shown  in  Fig.  35  ; the  two 
outer  lenses  are  of  crown  glass  and  convergent,  and 
the  middle  lens  is  of  flint  glass  and  divergent,  the  concave 
surfaces  being  all  turned  towards  the  incident  light. 

Let  A,  B,  C,  D,  as  in  the  figure,  be  the  points  in 
which  the  surfaces  are  cut  by  the  axis  of  the  system  : 
also  let 

H 


98 


PHOTOGEAPHIC  OPTICS 


o 


Fig.  35. 


ISTj  ^2  be  the  nodal  points  of  the  first  lens. 


N/Ng 


second  lens. 


V 


ELEMENTARY  THEORY  OF  LENSES  99 


]Sr/'  ^2"  be  the  nodal  points  of  the  third  lens. 

Ml  M2  „ „ „ combination  of 

1st  and  2nd 
lenses. 

Li  L2  „ jj  whole  combin- 

ation. 

These  points  as  determined  below  are  shown 
on  a magnified  scale  in  Fig.  36. 

To  give  the  calculations  in  full  would  occupy  too 
much  space,  so  the  summary  only  is  given. 

Let/,/',  r,  s,  e,  etc.,  have  the  meanings  assigned 
to  them  in  § 44,  then  : 

First  lens:  r = 4*29  inches,  s = 1*20  inch, 
refractive  index  = Pi  = 1*5146,  e = *230  inch  ; 
from  this  we  get 


Z 

rg 


= X = — *206  inch.  (§  44) 

B = y — — *057  inch. 

Fi  = focal  length  = — 3*159  inches. 


Second  lens:  r = 1*20  inch,  s = 3*75  inches, 
refractive  index  = = 1*574,  e — ‘050  inch  ; 

from  this  we  get 

B N/  = = *015  inch. 

C N2'  = y = ’047  inch. 

F2  = focal  length  = 3*095  inches. 


Third  lens  : r = 3*75  inches,  s = 1*80  inch,  ^ 
refractive  index  = = I'blT,  e = *151  inch; 

from  this  we  get 

CN/  = x"  = - *186  inch. 

D N2"  — y'  = — *089  inch. 

F3  = focal  length  = — 6*51  inches. 


Now  combine  the  first  and  second  lenses,  § 53,  we 
have  e = distance  between  N^  and  N^',  which  is  in 
the  negative  direction  (see  figure)  = — (B  N2  + 
B N/)  = - (-057  + *015)  = - *072  inch. 

Fj^  = — 3*158  inches,  F2  = 3*095  inches. 


cs/ 

-J 


Fie.  36. 


100 


PHOTOGRAPHIC  OPTICS 


Hence 

Ml  Ni 


-e-Pi  = 

« + ^1  + F2 

_ -072  X 3-158 
035 


1-68  inch. 


y = 


e P. 


2 _ 


(p  = focal  length  = 


e + F,  + F, 

Fil'2 


•072  X 3 095 
035 


l-65i 


« + Fi  + P2 

3-158  X 3-095 

T35 


= 71-76  inch. 


. - . Mj  A = 1-68  - -206  = 1-47  inch. 
MgC  = 1-65  + -05  = 1-70  inch. 

Now  add  on  the  third  lens,  we  have 

e = distance  between  Nj"  and  Mj. 

= -186  + 1-70  = 1-886  inch. 

(p  = 71-76  inches,  F3  = — 6-51  inches. 


L.  N " = 


e <p 


(P  + Fs  + e 

1-886  X 71-76 


e Fo 


67-14 

1-886  X 6-51 


(p  + F^+e  67-14 

F = focal  length  = = 

S (#)  + F3  + 6 

_ 71-76  X 6-51 

67-14 


= — 2-013  inches. 
= - -183  in. 


— 6-96  inches. 


. - . ALi  = - (Lj  Mj-AMj)  = 

- (2-013  - 1-478)  = - -535  inch. 

D Lg  = - (L,  N2"  - N/'  D)  = 

- (-183  + -089)  = - -272  inch. 

Thus  we  have  the  focal  length  of  the  combination,  and 
the  positions  of  its  nodal  points. 


ELEMENTARY  THEORY  OF  LENSES 


101 


This  example  is  enough  to  show  how  the  calcu- 
lations are  worked  ; the  case  in  which  the  lenses  are 
not  in  contact  will  only  differ  from  this  in  altering  the 
values  of  e,  the  distances  between  the  nodal  points,  and 
presents  no  difficulty. 

59.  Definitions  of  Certain  Terms. — In  the  present 
section,  definitions  will  be  given  of  several  angles 
connected  with  a lens. 

Let  us  consider  the  illumination  of  a plate  at  points 
at  varying  distances  from  the  axis  ; at  a point  on  the 
axis  the  aperture  appears  circular.  As  the  point  con- 
sidered recedes  from  the  axis  the  pencils  which  illumin- 
ate it  become  oblique,  and  the  aperture  is  foreshortened, 
in  consequence  of  which  the  illumination  decreases 
regularly,  and  its  rate  of  decrease  can  be  calculated  ; 
but  after  a certain  distance  the  light  begins  to  be  cut 
off  by  the  mounting,  and  if  we  go  far  enough  is  alto- 
gether intercepted. 

If  we  consider  the  whole  circumference,  we  can  then 
imagine  two  cones  having  their  vertices  at  the  nodal 
point  of  emergence. 

On  the  surface  of  the  first  cone,  called  the  cone  of 
illumination^  lie  the  axes  of  the  extreme  pencils,  any 
portion  of  which  escapes  the  mounting,  it  therefore 
contains  all  the  light  which  reaches  the  plate  ; on  the 
surface  of  the  second  cone  lie  the  axes  of  the  extreme 
pencils  which  are  not  at  all  cut  off  by  the  mounting,  this 
may  be  called  the  cone  outside  which  the  aperture  begins 
to  he  eclipsed. 

Inside  the  latter  cone  the  illumination  decreases 
regularly  and  not  very  rapidly  as  we  leave  the  axis,  but 
outside  it  the  decrease  is  irregular  and  generally  rapid; 
it  is  therefore  necessary  to  place  the  plate  inside  the 
inner  cone  to  obtain  anything  like  equal  exposure  all 
over. 

In  judging  of  a lens  it  is  therefore  important  to  know 
the  angles  of  these  cones  (§  115). 

The  angle  of  sharpness  is  the  angle  between  the 


102 


PHOTOGRAPHIC  OPTICS 


axes  of  the  extreme  pencils  which  can  be  focussed  on 
the  plate  to  the  sharpness  required  ; it  is  a matter  of 
every-day  experience  that  this  angle  varies  with  the 
stop  used,  for  the  usual  way  to  make  the  definition  sharp 
at  the  edge  of  a picture  is  to  reduce  the  stop. 

If  this  angle  is  known  for  a lens  with  a particular 
stop,  we  can  determine  how  large  a plate  the  lens  will 
cover  with  that  stop. 

Let  (Fig.  37)  N,  N'  be  the  nodal  points  of  incidence 
and  emergence,  x y the  axis  of  the  lens,  and  let  A N, 
N'  D and  B FT,  N'  C be  the  axes  of  the  extreme  pencils 
which  give  sharp  enough  images  with  the  stop  of  aperture 
E F;  for  landscape  work  the  plate  must  be  placed  either 
at,  or  very  near  to,  the  principal  focal  plane  of  the  lens, 
since  the  objects  are  distant. 

Let  F be  the  principal  focus,  and  let  'N'  C and  W D 
meet  the  focal  plane  in  C and  D ; then  C D will  be  the 
greatest  dimension  of  the  largest  plate  that  can  be 
used. 

The  angle  A N B (or  C N'  D which  is  equal  to  it) 
is  the  angle  of  sharpness  ; denote  this  by  2 6 degrees, 
and  let  F be  the  focal  length  of  the  lens,  then 

^ = CN'F  = tan  e,  .■.CF  = N'F  tan  6. 
N'F 

. • . C D = 2 C F = 2 N'  F 0 = 2 F i!aw 
C D can  therefore  be  found  by  the  aid  of  a table  of 
tangents  when  0 and  F are  known. 

If  we  do  not  wish  to  use  the  lens  for  landscape  work, 
but  for  copying,  the  plate  will  no  longer  be  at  distance 
F from  the  lens  ; we  must  in  this  case  find  the  distance 
from  N'  (call  it  v)  at  which  the  plate  is  placed,  and  we 
evidently  get 

CT>  = 2 V tan  6. 

It  is,  however,  possible  that  6 may  not  be  the  same 
in  this  latter  case  ; whether  it  is  so  or  not  can  be  found 
only  by  experiment. 

We  must  next  consider  what  the  length  C D is ; it  is 


ELEMENTARY  THEORY  OF  LENSES 


103 


the  diameter  of  a circle  within  which  everything  is  as 
sharp  as  required. 


104 


PHOTOGRAPHIC  OPTICS 


If  any  particular  sized  plate  will  go  inside  that  circle 
the  lens  will  cover  it  sharply. 

Thus  (Fig.  38)  the  plate  A C B D will  just  fit  inside 
the  circle  of  which  C D is  the  diameter,  and  C D is 
therefore  the  diagonal  of  the  plate. 

If  we  know  the  dimensions  of  the  plate  we  can  find 
its  diagonal,  for 


Fig.  38. 


Square  on  diagonal  = sum  of  squares  on  the  sides. 
Example. — Will  a lens  of  six  inches  focal  length, 
whose  angle  of  sharpness  is  fifty  degrees  with  its  largest 
stop,  cover  a plate  3^  X 4^  inches  h 

Here  diagonal  = ^ (3*25^  + 4 ’2 5^)  = 

J 28*62  = 5*35  inches. 
And  20  = 50°,  . * . d = 25° 

. ' . QT>  = 2 ¥ tan  0 = 2 X 6x  ta7i  25°  = 

12  X *4663  = 5*595  inches. 
Hence  C H,  being  greater  than  the  diagonal,  the 
lens  will  cover  the  plate  sharply. 


ELEMENTARY  THEORY  OF  LENSES 


105 


The  angle  of  view  of  the  lens  is  the  angle  between 
the  axes  of  the  two  extreme  pencils  of  rays  which  strike 
the  plate.  This  is  the  same  as  the  angle  between  the 
lines  joining  the  nodal  point  of  incidence  to  the  most 
widely  separated  objects  that  will  be  included  in  the 
picture. 

There  is  here  a danger  of  confusion,  for  we  may  take 
the  most  widely  separated  pencils  to  be  those  which 
strike  the  plate  at  the  extremities  of  a diagonal,  or  else 
those  which  strike  it  at  the  extremities  of  either  a 
horizontal  or  vertical  line.  We  shall  take  the  angle 
between  the  pencils  which  meet  the  plate  at  the  extre- 
mities of  a horizontal  line,  but  the  reader  should  be  on 
his  guard,  as  makers  in  their  catalogues  often  use  the 
diagonal.  The  angle  will  of  course  depend  on  which 
side  of  the  plate  is  horizontal.  For  purposes  of  com- 
parison it  will  be  well  to  keep  to  the  case  when  the 
longer  side  is  horizontal. 

To  find  the  angle  of  view,  let  CD  (Fig.  37)  represent 
the  longest  side  of  the  plate,  which,  in  landscape  work, 
is  at  the  principal  focus. 

Let  2 (/)  be  the  angle  of  view,  then 


C N'  D = 2 (j)  and  tan  (j)  = 


CF 

N'F 


CF  _ CD 
F 2F 


If,  therefore,  C F and  F are  known  we  can  calculate 
tan  (py  and  hence  find  (p  from  the  tables. 

Example. — Find  the  angle  of  view  of  a lens  of  six 
inches  focal  length  used  with  a plate  3 J X inches. 
Here  F =6  CD  = 4*25  CF  = 2T25. 

6 

.-.  cp  = 19°  30'  and  2 ^ - 39° 

Hence  the  angle  of  view  is  39°. 

59a.  Size  of  Image.  — When  a lens  is  used  for 
such  work  as  reducing  and  enlarging,  in  which  it  is 
comparatively  close  to  the  object  to  be  photographed. 


106 


PHOTOGRAPHIC  OPTICS 


the  relative  sizes  of  object  and  image  can  be  varied  by 
varying  their  distances  from  the  lens ; and  this  can  be 
done  in  many  cases  out  of  doors,  the  camera  being 
moved  till  the  picture  on  the  ground  glass  is  of  the 
right  size. 

But  in  landscape  work,  specially  where  the  scene  is 
fairly  distant,  this  cannot  be  done,  for  often  the  picture 
can  be  obtained  from  one  point  of  view  only ; neither 
can  we  adjust  the  size  of  the  picture  by  moving  the 
focussing  screen,  for  that  would  throw  the  picture  out 
of  focus.  The  only  adjustment  available  is  that  of 
changing  the  lens  and  using  one  of  a different  focal 
length.  By  doing  this  we  can  vary  the  distance  of  the 
plate  from  the  lens  and  yet  keep  the  picture  in  focus. 

The  shorter  the  focus  of  the  lens  the  nearer  is  the 
plate  to  it,  and  hence  the  larger  the  angle  of  view ; 
also  the  larger  the  region  whose  picture  is  included  on 
the  plate  and  the  smaller  is  any  particular  object.  If 
the  lens  is  of  long  focus,  and  the  angle  of  view  small, 
the  smaller  will  be  the  region  pictured,  and  the  larger 
any  particular  object. 

Hence  to  increase  the  size  of  the  image  of  a particular 
object  we  increase  the  focal  length  of  the  lens  used ; 
this  proceeding  is  limited  only  by  the  possible  extension 
of  the  camera. 

When  very  distant  objects  are  to  be  photographed  a 
lens  of  very  long  focus  must  be  used.  The  difficulty  of 
the  extension  of  the  camera  has  been  met  by  the  use  of 
lenses  of  a special  design  called  “ telephotographic  ” ; 
one  of  the  best  known  of  these  is  that  of  Dallmeyer, 
and  as  it  furnishes  an  excellent  example  of  the  prin- 
ciples of  lens  combination  we  shall  describe  it  in  the 
next  section. 

60.  Dallmeyer’s  Telephotographic  Lens.  — This 
objective  consists  of  a converging  lens  or  combination 
in  front,  and  a diverging  lens  or  combination  behind  ; 
for  simplicity  these  will  here  be  represented  by  single 
lenses. 


ELEMENTARY  THEORY  OF  LENSES 


107 


It  is  not  hard  to  see  that 
this  arrangement  can  be  made  to 
act  as  a lens  of  great  focal 
length  and  small  angle  of  view. 
For  (Fig.  39)  let  A be  the  con- 
verging lens,  parallel  rays  falling 
on  this  would  ordinarily  con- 
verge to  the  point  P ; but  they 
are  made  to  fall  on  the  diverging 
lens  B,  which  causes  them  to 
converge  to  a point  F much 
further  back,  where  F is  the 
image  of  P produced  by  the 
second  lens.  The  rays  which 
reach  F are  therefore  inclined  as 
if  they  came  from  some  lens  N 
very  much  further  away  than 
either  A or  B. 

Thus  the  combination  is  equiva- 
lent to  a lens  of  very  much 
greater  focal  length  than  that 
of  either  of  the  component  lenses, 
and  thus  gives  us  the  means 
of  obtaining  by  the  aid  of  lenses 
comparatively  near  the  plate  the 
effect  of  a lens  placed  at  a much 
greater  distance  away. 

To  calculate  the  exact  effect 
of  any  particular  combination  we 
can  use  the  formulae  we  have 
already  obtained,  i.  e.  (§  53) — 

— -^1-,  y = 

^ + /l  + /2  + /l  + A 

F = A/2 

^ + /i  + A 

where  e,  etc.,  have  the  mean- 
ings already  assigned. 


; 2 : 


o 

Fig.  39. 


/ 


Example  (Fig.  40). — Let  f\  — — 6 inches,  ^2  “ ^ 
inches,  then 


ELEMENTARY  THEORY  OF  LENSES 


109 


and  we  can  make  e to  have  any  value  that  may  be 
convenient. 

The  equivalent  lens  must  be  convergent,  and  hence 
F negative  or  6 — 3 positive,  e > 3. 

If  e = 3 then  x = — ^ y = +oo,F=  — oo, 
or  the  combination  produces  no  effect. 

Take,  for  example,  e = 3f  inches,  then  we  get  x — 
30  inches,  y = 15  inches,  F = — 24  inches.  The 
nodal  point  of  emergence  of  the  equivalent  lens  is  there- 
fore fifteen  inches  in  front  of  the  nodal  point  of 
emergence  of  the  diverging  lens,  and  its  focal  length  is 
twenty-four  inches.  The  lengths  in  this  case  are  shown 
on  Fig.  40,  adapted  from  that  in  Dallmeyer’s  pamphlet, 
and  will  repay  careful  study.  The  position  of  one  only  of 
the  nodal  points  is  shown,  the  other  being  beyond  the 
limits  of  the  figure. 

Two  sets  of  oblique  pencils  of  rays  are  shown ; the 
first  pair  are  near  enough  to  the  axis  to  fall  on  the 
second  lens  and  be  focussed  on  the  plate,  but  the  second 
pair  do  not  strike  the  lens  at  all,  but  are  absorbed  by 
the  mount  of  the  diverging  lens. 

By  varying  the  distance  e we  can  vary  the  focal 
length  of  the  combination,  and  could  make  it  as  great 
as  we  like  were  it  not  that  a practical  limit  is  placed  by 
the  possible  extension  of  the  camera. 

If  we  suppose  the  diverging  lens  rigidly  fixed  to  the 
camera,  while  the  adjustment  of  e is  made  by  moving 
the  front  lens,  we  can,  when  we  know  the  possible 
extension,  find  the  greatest  focal  length  obtainable. 

Example. — With  the  lens  of  the  last  example,  if  the 
camera  can  be  extended  till  the  ground  glass  is  twelve 
inches  distant  from  the  diverging  lens,  find  the  distance 
e between  the  lenses  and  the  corresponding  focal  length. 

The  distance  between  the  plate  and  the  back  lens  is 
equal  to  the  difference  between  the  focal  length  of  the 
combination  and  the  distance  of  the  back  lens  from  the 
nodal  point  of  emergence  of  the  combination  F — 
Distance  between  plate  and  back  lens 


110 


PHOTOGRAPHIC  OPTICS 


= - Y - y 


e/2 


- /1/2 

e +/i  4-/2  e +yi  -I-/2 

__  /2  (e  + /i) 


In  the  present  case  this  becomes 

12  = - ^ 

e - 3 


« +/i  4-/2 


This  is  a simple  equation  for  e,  and  when  solved  gives 
us  e = 34  inches.  From  this  we  can  find  that  F = — 
30  inches.  Hence  thirty  inches  is  the  greatest  focal 
length  that  can  be  obtained  with  the  given  camera. 

61.  Angle  of  View  of  Telephotographic  Lens. — The 
angle  of  view  can  of  course  be  found  as  before  when  the 
focal  length  used  is  known. 

Example. — Find  the  angle  of  view  of  the  lens  adjusted 
as  in  the  first  example  of  the  last  section  when  used 
with  a plate  X 3^  inches. 

Here  F = - 24,  CD  = Pi  (§59),  CF  = 2T25 


.*.  tan  0 


CF 

”F 


2T25 

”24~~ 


•0886 


d = 5°  4'  .-.  2d  = 10°  8' 

.*.  Angle  of  view  = 10°  8' 

62.  Magnification. — Dallmeyer  reckons  as  the  mag- 
nification, the  ratio  of  the  size  of  image  of  a distant 
object  produced  by  the  compound  lens  to  that  of  the 
image  produced  by  the  converging  lens  alone. 

The  advantage  of  this  proceeding  is  that  we  use  the 
image  produced  by  the  converging  lens  alone  as  the 
standard  with  which  we  compare  the  size  of  the  image 
produced  by  the  combination. 

To  find  the  ratio,  we  notice  first  that  the  distant 
object  in  question  will  subtend  angles  at  the  nodal 
points  of  incidence  of  the  converging  and  of  the 
equivalent  lens  which  are  practically  equal,  and  hence 
the  angles  subtended  by  the  images  at  the  nodal  points 
of  emergence  of  their  respective  lenses  will  be  equal. 


ELEMENTARY  THEORY  OF  LENSES 


111 


Let  L and  M,  Fig.  41,  be  the  nodal  points  of  emergence 
of  the  equivalent  and  converging  lenses  respectively,  and 
let  A F B,  C F D be  the  corresponding  images,  F being 
the  point  where  the  axis  of  the  lens  meets  the  image; 
then 


Fig.  41. 


Size  of  image  by  combination  ^F 

Size  of  image  by  converging  lens  CD  C F M F 
_ focal  length  of  combination, 
focal  length  of  converging  lens. 

For  since  the  angles  ALB,  C M D are  equal,  triangles 
ALB,  C M D will  be  similar,  and  so  also  will  be  the 


112 


PHOTOGRAPHIC  OPTICS 


triangles  A L F,  C M F.  We  see,  therefore,  that  the 
magnification  is  expressed  by  the  ratio  of  the  focal 
lengths. 

In  the  case  considered  above  this  ratio  is  24/6  = 4, 
and  the  size  of  the  image  produced  by  the  combination 
is  four  times  that  produced  by  the  converging  lens 
alone. 

It  should  be  remarked  that  no  attempt  has  been 
made  to  make  the  notation  of  these  sections  correspond 
with  that  used  by  Dailmeyer,  nor  have  either  general 
expressions  or  rules  been  given.  Any  question  that 
may  arise  can  easily  be  treated  when  the  general  prin- 
ciples of  the  combination  are  known  ; rules  tend  only 
to  produce  confusion. 

63.  Perspective. — Photographs  of  buildings  when 
taken  with  a wide-angle  lens  often  present  a strained 
or  distorted  appearance,  which  is  due  to  the  fact  that 
the  distances  between  the  various  points  in  the  photo- 
graph do  not  subtend  at  the  eye  the  same  angles  as  the 
distances  between  the  corresponding  points  in  the  object 
photographed,  or,  in  other  words,  the  perspective  of  the 
photograph  is  not  correct.  This  defect  is  particularly 
noticeable  when  the  photograph  is  taken  with  a lens  of 
very  short  focal  length ; if,  however,  an  enlargement  of 
such  a picture  is  made  its  appearance  is  usually  very 
much  better  than  that  of  the  original  picture. 

To  explain  this,  let  A B,  C D (Fig.  42)  be  distant 
objects,  and  let  ah,  cdh^  the  corresponding  images ; if 
ISTj  ^2  be  the  nodal  points  of  incidence  and  emergence 
respectively,  the  lines  joining  these  points  to  the  corre- 
sponding points  of  object  and  .image  are  parallel — for 
instance,  A a Ng  are  parallel.  Thus  the  angles 
A Nj  B,  are  equal,  and  so  also  are  the  angles 

C ]Sr^  D,  c N2  d,  or  the  angles  subtended  by  the  images 
at  the  nodal  point  of  emergence  are  equal  to  those  sub- 
tended by  the  corresponding  points  of  the  object  at  the 
nodal  points  of  incidence. 

If  then  the  eye  be  placed  opposite  the  middle  of  the 


ELEMENTARY  THEORY  OF  LENSES 


113 


picture  and  at  a distance  from  it  equal  to  N2  F,  the 
various  parts  will  subtend  at  the  eye  angles  equal  to 
those  subtended  by  the  corresponding  parts  of  the 

I 


114 


PHOTOGRAPHIC  OPTICS 


object,  and  the  perspective  of  the  picture  will  be 
correct.  But  in  many  cases  the  distance  N2  F is  less 
than  the  distance  of  distinct  vision  for  normal  eyes, 
and  to  see  the  picture  at  all  it  must  be  held  at  a dis- 
tance greater  than  N2  F from  the  eye,  such,  for  instance, 
as  P F. 

The  angles  which  a P 5,  cT  d now  subtend  at  the  eye 
are  less  than  the  angles  a ^2  6,  0X26^  (which  we  have 
seen  are  the  correct  angles),  and  distortion  will  therefore 
result ; the  picture  will  appear  crowded  into  a smaller 
space  than  it  ought  to  occupy.  And  it  can  be  shown 
that  the  angles  between  lines  at  the  edges  of  the  picture 
will  appear  diminished  if  they  are  acute  and  increased 
if  they  are  obtuse ; on  both  these  accounts  the  picture 
will  not  appear  to  have  the  proper  perspective. 

Now,  suppose  that  an  enlargement  is  made — the 
effect  of  this  is  to  increase  all  the  lengths  in  the  picture 
proportionally — thus  if  the  picture  be  enlarged  three 
times,  it  will  be  exactly  similar  to  a picture  of  the  same 
object  taken  with  a lens  of  focal  length  three  times 
that  of  the  lens  originally  used.  The  distance  N2  F 
from  the  picture  of  the  point  where  the  eye  should  be 
placed  to  get  the  proper  perspective  is  three  times  as 
great  for  the  enlargement  as  for  the  original  picture. 
If  it  is  greater  than  the  least  distance  of  distinct  vision 
the  picture  can  be  seen  by  an  eye  placed  at  that  dis- 
tance. This  explains  why  an  enlargement  is  often 
more  pleasing  than  the  original  picture. 

The  least  distance  of  distinct  vision  for  a normal 
eye  is  somewhere  about  ten  inches  ; it  follows  therefore 
that  a picture  to  have  the  proper  perspective  must  be 
taken  with  a lens  whose  focal  length  is  greater  than 
ten  inches.  A telephotographic  lens,  the  focal  length 
of  which  can  be  as  much  as  five  feet,  will  obviously 
produce  pictures  with  much  better  perspective  than  an 
ordinary  lens. 

It  follows,  therefore,  that  for  each  picture  there  is  a 
definite  position  from  which  it  should  be  viewed  if  the 


ELEMENTARY  THEORY  OF  LENSES 


115 


proper  effect  is  to  be  obtained,  i.e.  a point  on  a line  at 
right  angles  to  the  picture  opposite  to  its  centre  and  at 
a distance  from  it  equal  to  the  focal  length  of  the  lens 
with  which  it  was  (or  in  the  case  of  an  enlargement 
with  which  it  could  have  been)  taken. 

64.  The  Use  of  the  Swing  Back. — A defect  in  the 
picture  which  must  not  be  confounded  with  that  of  tlie 
last  article  is  caused  by  the  plate  being  placed  in  a 
wrong  position.  When  arranging  the  camera  it  is  often 
necessary  to  give  it  a tilt  (to  get  on  the  plate  all  that 
is  required),  which  tilts  the  plate  also  ; if  the  photo- 
graph is  taken  with  the  plate  in  that  position  a certain 
kind  of  distortion  is  produced.  If  the  picture  is  a 
landscape  with  no  near  or  large  buildings  in  it  the 
distortion  is  not  noticeable,  but  if  it  contains  parallel 
straight  lines  such  as  occur  in  buildings  or  diagrams, 
these  straight  lines  will  in  the  picture  run  together  at 
one  end  or  another  according  to  the  way  in  which  the 
camera  is  tilted,  giving  to  buildings  the  appearance  of 
tumbling  down. 

To  understand  the  cause  of  this  running  together  let 
the  object  be  a rectangle  A B C D (Fig.  43),  and  let 

N2  be  the  nodal  points  of  incidence  and  emergence 
respectively ; join  to  A B C D,  and  let  N2  a,  ^2 
N2  c,  N2  c?  be  lines  through  the  other  nodal  point  parallel 
to  the  lines  through  the  first.  Suppose  the  plate  to  be 
parallel  to  A B C D and  to  cut  the  lines  through  N2  in 
the  points  ah  c d so  that  ah  c d is  the  image  of 
ABCD. 

Then,  by  similar  figures,  since  A B and  C D are 
equal  and  parallel,  a h and  c d are  also  equal  and 
parallel ; thus  the  lines  a d,  c h appear  parallel  in  the 
picture  as  they  should  do. 

If,  however,  the  plate  be  tilted  into  the  position 
a'  h'  c d j where  a V and  c dt  are  parallel  to  a h^  c d 
respectively,  the  picture  may  still  be  fairly  in  focus,  but 
c d!  is  now  longer  than  c d and  ah’  is  shorter  than 
a 6,  which  shows  that  c d!  is  now  greater  than  a!  h\ 


116 


PHOTOGRAPHIC  OPTICS 


Fia.  43. 


ELEMENTARY  THEORY  OF  LENSES 


117 


The  effect  of  this  is  that  the 
lines  a d\  c h'  in  the  picture 
are  no  longer  parallel,  and  a 
certain  kind  of  distortion  re- 
sults. 

We  have  here  supposed  the 
plate  to  be  tilted  about  a hori- 
zontal axis,  but  if  it  be  tilted 
about  a vertical  axis  it  is 
evident  that  the  images  of  the 
lines  A B,  C D will  no  longer 
be  parallel. 

We  see  thus  that  if  the 
photograph  is  to  reproduce 
parallel  straight  lines  correctly, 
the  plate  must  be  placed  with 
its  plane  parallel  to  that  of 
the  object ; for  this  purpose 
cameras  are  now  supplied  with 
an  arrangement  called  a swing 
back,  which  makes  it  possible 
to  keep  the  plane  vertical  and 
parallel  to  the  object,  however 
(within  certain  definite  limits) 
the  camera  may  be  placed. 
The  swing  about  a horizontal 
axis  is  that  most  frequently 
wanted,  but  the  swing  about  a 
vertical  axis  cannot  be  dis- 
pensed with  when  buildings 
are  photographed  from  awkward 
positions. 

In  landscape  work  a little 
distortion  of  the  kind  men- 
tioned is  not  usually  noticeable, 
and  the  swing  back  may  in 
consequence  be  put  to  quite 
a difierent  use  from  that  for 


118 


PHOTOGRAPHIC  OPTICS 


which  it  was  designed.  If  a 
picture  includes  objects  at  very 
different  distances  it  is  im- 
possible to  get  both  near  and 
far  objects  all  into  focus  at 
once.  Take,  for  example,  a 
photographer  on  the  top  of  a 
sea  cliff  who  wants  to  photo- 
graph a number  of  ships  stretch- 
ing away  from  close  beneath 
him  to  a considerable  distance. 

Let  A and  B (Fig.  44)  be 
two  of  the  ships,  and  let  a 
and  h be  the  images  of  these  ] 
then  if  A be  nearer  than  B,  a 
will  be  further  from  the  lens 
than  6,  or  the  line  a h will  be 
inclined  to  the  axis  of  the  lens. 
The  best  focussing  over  the 
whole  picture  can  therefore  be 
obtained  by  placing  the  plate 
so  that  both  a and  h are  on  it, 
which  can  be  done  by  the  aid 
of  the  swing  back.  A similar 
adjustment  can  sometimes  be 
made  by  the  swing  about  a 
vertical  axis. 

65.  A Property  of  the  Nodal 
Point  of  Emergence  and  Pano- 
ramic Photography.  — The 
nodal  point  of  emergence 
possesses  a property  which  is 
very  useful  both  in  lens  testing 
and  panoramic  photography. 
If  a lens  be  pivoted  to  turn 
about  an  axis  at  right  angles 
to  the  axis  of  the  lens,  pass- 
ing through  the  nodal  point 


Fig.  45. 


ELEMENTARY  THEORY  OF  LENSES 


119 


of  emergence,  and  the  picture  of 
a distant  object  formed  by  it  on 
a ground  glass  screen  be  observed, 
the  position  of  the  picture  on 
the  screen  will  remain  station-  jj 
ary  while  the  lens  is  revolved.  ' 

To  prove  this  let  and  N2 
(Fig.  45  a)  be  the  nodal  points 
of  incidence  and  emergence  re- 
spectively, and  let  the  lens  be 
pivoted  about  an  axis  through 
N2  perpendicular  to  the  axis  of 
the  lens ; let  be  the  line 

joining  a point  of  the  distant 
object  to  Nj,  and  L2  1^2  the  line 
joining  the  corresponding  point 
of  the  image  to  N2,  then  we 
know  (§44)  that  and  L2 

1^2  parallel. 

Now  let  the  lens  be  rotated 
through  any  angle  about  N2  till 
Ni  comes  to  N/,  the  line  N/  L 
joining  N^'  to  the  same  point  of 
the  object  will,  since  the  object 
is  distant,  be  parallel  to  N^  L^. 

The  line  joining  N2  to  the  corre- 
sponding points  of  the  image 
will  be  parallel  to  N^'  L and 
hence  to  N2  L2,  and  it  has  thus 
not  altered  its  position. 

That  this  is  not  true  when 
the  lens  is  rotated  about  any 
other  axis  than  that  stated  will 
be  evident  from  an  inspection  of  Fig.  4.5a. 

Fig.  45a. 

Hence  the  image  of  the  point  in  question  will  be  in 
the  same  position  whatever  the  position  of  the  lens,  and 
this  is  true  of  all  points  of  the  image.  Thus  as  the 


120 


PHOTOGRAPHIC  OPTICS 


lens  is  rotated  the  picture  remains  stationary,  but  is 
extinguished  at  one  side  and  extended  at  the  other, 
very  much  as  if  a long  map  on  rollers  is  laid  on  a 
table,  and  one  side  is  rolled  up  while  the  other  is 
unrolled  without  sliding  the  map  along  the  table. 

This  principle  has  been  applied  both  in  England  and 
France  to  the  construction  of  a panoramic  camera  : the 
essential  parts  of  such  an  apparatus  are,  firstly,  a lens 
pivoted  as  described,  and  secondly,  a sensitive  film 


Fig.  46. 


arranged  in  the  form  of  a semi-cylinder  with  the  axis 
of  rotation  for  axis.  To  obtain  a uniform  exposure  all 
along  the  roll  of  film,  clockwork  has  been  used  to  rotate 
the  lens  uniformly. 

A rectilinear  lens  must  be  employed,  as  other  lenses 
have  a distortion  at  the  edges  (to  be  described  later) 
which  would  cause  a slight  shifting  at  the  edges  of  the 
pictures  as  the  lens  revolves. 

Fig.  46  is  a camera,  designed  by  M.  A.  Moessard,  for 


ELEMENTARY  THEORY  OF  LENSES 


121 


panoramic  work.  A camera  of  a similar  nature  has 
been  designed  by  Col.  Stewart,  R.E. ; the  main  differ- 
ence being  that  the  film  instead  of  being  in  the  form  of 
a cylinder  is  wound  on  rollers  and  suitably  unrolled  as 
the  carhera  revolves. 


CHAPTER  III 

ABERRATION 

65a.  Introductory. — We  have  now  to  see  how  the 
theory  given  in  the  last  chapter  must  be  modified  to 
express  the  real  state  of  affairs  when  we  come  to  actual 
practice.  To  simplify  the  work  we  made  several 
assumptions  which  are  not  altogether  true ; they  were 
as  follows  : 

(а)  All  the  pencils  of  light  dealt  with  were  slender, 
no  ray  of  light  being  far  from  any  other  ray. 

(б)  All  pencils  of  light  were  incident  near  the 
centre  of  the  lens,  and  thus  only  a small  portion 
of  each  surface  was  used. 

(c)  The  axes  of  the  incident  pencils  were  inclined 
at  only  a small  angle  to  the  axis  of  the  lens. 

{(T)  The  light  was  supposed  to  be  monochromatic, 
so  that  a single  ray  gave  rise  to  only  one 
refracted  ray. 

That  these  assumptions  are  not  strictly  true  is  a 
matter  of  common  experience.  The  aperture  of  the 
lens  is  often  by  no  means  small,  and  the  incident  pencil 
of  light  is  in  consequence  not  slender ; besides  this, 
even  in  landscape  work,  where  lenses  of  a moderate 
angle  of  view  are  mostly  used,  the  axes  of  the  extreme 
pencils  are  often  inclined  at  an  angle  of  30°  to  the 
axis  of  the  lens,  an  angle  which  cannot  be  called  small ; 
lastly,  photographic  work  is  done  by  the  aid  of  daylight 
or  lamplight,  neither  of  which  is  even  approximately 

122 


ABERRATION 


123 


monochromatic.  Why  then  have  we  taken  so  much 
trouble  to  investigate  an  imaginary  state  of  affairs  ? 

It  is  because  this  ideal  state  leads  us  to  a very  fair 
approximation,  from  which,  by  the  addition  of  small 
corrections,  we  can  deduce  a more  nearly  accurate 
statement  of  the  case ; and  also  because  we  can,  by 
using  two  or  more  lenses,  approximate  very  closely  to 
the  ideal  state. 

The  subject  is  usually  divided  into  two  parts  : firstly, 
that  of  Spherical  Aberration,  due  to  the  use  of  large 
pencils  or  of  pencils  oblique  to  the  axis  of  the  lens ; 
secondly.  Chromatic  Aberration,  due  to  the  various 
coloured  rays  in  the  incident  light.  These  two  kinds 
of  aberration  will  be  treated  in  turn,  and  we  shall  see 
how  they  modify  the  relations  found  in  the  last  chapter 
without  altering  their  general  character. 

The  subject  of  aberration  is  undoubtedly  more  diffi- 
cult than  the  elementary  theory  of  a lens,  not  because 
the  fundamental  ideas  are  very  difficult,  but  because  the 
calculations  are  unavoidably  complicated.  The  practical 
photographer  is  not  much  concerned  with  aberration, 
for  generally  he  is  content  to  use  the  lens  which  the 
study  and  skill  of  the  maker  have  produced,  and  to  ask 
no  questions  provided  the  results  are  satisfactory ; 
besides  this,  the  production  of  a good  lens  requires  so 
much  skill  and  accuracy  of  work,  that  very  few 
amateurs  could  hope  to  produce  one  worth  using.  The 
calculations  necessary  for  designing  lenses  are  long  and 
arduous,  not  so  much  on  account  of  the  intricacy  of  the 
principles  involved,  but  on  account  of  the  complicated 
nature  of  the  algebra  required  to  find  the  magnitudes 
of  the  various  quantities. 

It  is  doubtful  whether  these  calculations  are  per- 
formed except  by  lens  designers  accustomed  by  practice 
to  such  work ; or  whether  it  is  profitable  or  even 
desirable  for  the  general  reader  to  attempt  them. 

But  though  the  numerical  calculations  are  difficult,  a 
knowledge  of  general  principles  is  of  great  use  for  the 


124 


PHOTOGRAPHIC  OPTICS 


proper  understanding  of  the  general  character  of  a lens 
and  in  estimating  and  testing  its  capabilities. 

We  shall  therefore  give  in  the  first  place  a minute 
description  of  the  action  of  a lens,  and  after  this  the 
formulae  usually  quoted  in  optical  treatises,  but,  except 
in  a few  cases,  we  shall  not  enter  on  the  demonstration 
of  them. 

Those  readers  who  wish  for  further  information 
should  consult  Coddington’s  or  Wallon’s  L^Ohjectif 

Photographique.  A great  deal  of  interesting  informa- 
tion and  specimens  of  calculations  for  lens  designing 
are  given  in  a paper  by  M.  Martin  ^ on  the  ‘ Determina- 
tion des  courbures  des  objectifs.’ 


I. — Spherical  Aberration. 

656.  Let  us  take  first  the  case  of  a pencil  of  light 
emanating  from  a point  on  the  axis  of  a lens  and  strik- 
ing the  lens  symmetrically.  Such  a pencil  can  be 
produced  by  placing  a screen  pierced  with  a small  hole 
in  front  of  a gas  flame,  the  hole  being  on  the  axis  of 
the  lens.  The  pencil  is  here  not  supposed  small  as  in  the 
previous  chapter. 

If  a white  screen  be  placed  on  the  side  of  the  lens 
away  from  the  light,  and  be  moved  backwards  and  for- 
wards, and  kept  paraded  to  the  lens,  the  phenomena 
presented  will  be  as  follows. 

When  the  screen  is  very  close  to  the  lens  the  appear- 
ance on  it  is  a circle  of  light  uniformly  illuminated  ; 
when  the  screen  moves  away  from  the  lens  the  circle  of 
light  contracts,  and  becomes  brighter  at  the  edge  than 
at  the  centre.  As  the  movement  of  the  screen  is 
continued  the  circle  contracts  further  and  the  edge 
becomes  still  brighter,  and,  if  looked  for  carefully,  a 

^ Amiales  de  VEcole  normctle  sup^rieure,  1877.  Gauthier- Villars, 
Paris. 


ABERRATION 


125 


brighter  spot  at  the  centre  can  also  be  seen.  After 
this  the  bright  ring  shrinks  till  it  becomes  a patch. 

Up  to  this  the  space  outside  the  illuminated  circle 
has  been  dark ; but  just  about  when  the  bright  ring 
becomes  a patch,  the  space  outside  it  becomes  diffusely 
illuminated. 

When  the  screen  is  far  enough  back  the  bright 
patch  constitutes  the  image  of  the  luminous  point,  an 
image  of  the  hole  through  which  the  light  is  admitted 
being  formed ; and  further  back  still  the  patch  becomes 
indefinite  and  grows  fainter,  while  the  circle  of  diffuse 
light  round  it  increases  rapidly  in  size. 

It  has  proved  impossible  to  get  really  successful 
photographs  of  these  phenomena,  owing  mainly  to 
diffuse  light  which  cannot  be  got  rid  of,  but  the 
phenomena  themselves  can  easily  be  reproduced  by 
any  one  possessing  a lens  of  fair  size,  such  as  a magnify- 
ing glass  or  one  of  the  condensing  lenses  of  an  enlarging 
lantern ; since  the  object  is  to  get  as  much  aberration 
as  possible  an  uncorrected  lens  should  be  used. 

The  incident  light  should  be  rendered  monochromatic 
by  interposing  a coloured  glass  between  the  luminous 
point  and  the  lens. 

The  various  appearances  described  are  all  of  them 
sections,  by  planes  perpendicular  to  the  axis  of  the  lens, 
of  the  assemblage  of  the  rays  of  light  after  refraction 
by  the  lens. 

Let  us  proceed  to  form  some  idea  of  the  nature  of 
the  assemblage  which  gives  rise  to  these  sections.  The 
arrangement  of  rays  which  will  be  found  to  answer  all 
requirements  is  given  in  Fig.  47,  which  represents  a 
section  by  a plane  passing  through  the  axis  of  the 
lens. 

[In  the  figure  the  incident  rays  are,  for  a subsequent 
purpose,  taken  to  be  parallel  to  the  axis  of  the  lens, 
but  the  general  character  of  the  phenomenon  is  the 
same  as  when  the  rays  proceed  from  a point  on  the 
axis.] 


126 


PHOTOGRAPHIC  OPTICS 


It  should  be  noticed  that  the  rays  from  the  centre  of 
the  lens  are  shown  as  converging  to  F,  which  is  there- 
fore the  focus  conjugate  to  the  luminous  point ; while 


those  refracted  through  the  edges  converge  to  A,  a 
point  nearer  to  the  lens  than  F.  Rays  through  other 
portions  of  the  lens  cut  the  axis  in  points  lying  between 
A and  F. 


ABERRATION 


127 


At  a certain  distance  from  the  lens  (beyond  H and 
K in  the  figure)  the  rays  intersect,  and  there  is  between 
H,  K and  F a curve  formed  by  the  intersections  of  con- 
secutive rays.  Since  through  each  point  of  this  curve 
more  than  one  ray  passes,  it  follows  that  the  illumination 
there  is  brighter  than  at  points  inside  it ; or  in  other 
words,  the  curve  is  one  of  maximum  illumination.  This 
curve  is  called  a caustic  curve,  and  the  surface  of  which 
it  is  the  section  a caustic  surface.  The  points  on  the 
axis  lying  between  A and  F have  also  several  rays 
passing  through  each  of  them,  and  the  line  A F is 
therefore  one  of  maximum  illumination  and  may  be 
reckoned  as  part  of  the  caustic. 

That  this  agrees  with  observation  will  be  evident  if 
the  section  by  a screen,  parallel  to  the  lens  at  different 
distances  from  it,  be  considered.  At  C D the  section 
by  the  screen  is  evidently  of  a minimum  size ; beyond 
C D the  rays  which  converged  to  A spread  out  beyond 
the  caustic  surface,  producing  a bright  circle  of  light 
surrounded  by  diffused  light. 

Careful  examination  shows  that  rays  from  a con- 
siderable portion  of  the  lens  converge  to  F,  making  it 
the  most  brightly  illuminated  point,  so  that  the  image 
formed  at  F is  much  brighter  than  that  formed  at  any 
other  point  between  A and  F,  and  will  stand  out  clearly 
in  spite  of  the  diffuse  light  surrounding  it. 

Beyond  F there  is  no  point  where  the  rays  intersect, 
and  consequently  no  point  of  maximum  illumination, 
though  for  some  distance  the  patch  of  light  on  the 
screen  will  be  brightest  at  the  centre  fading  away 
gradually  to  the  edge. 

This  description  may  be  still  further  verified  by  first 
placing  in  front  of  the  lens  and  close  to  it  a screen  with 
a small  hole  cut  in  it  which  allows  rays  to  strike  only 
the  central  portions  of  the  lens,  when  the  focus  will  be 
found  to  be  at  F ; and  afterwards  removing  the  screen 
and  placing  a circular  disc  in  front  of  the  lens  which 
cuts  off  all  the  rays  except  those  near  the  edges,  when 


128 


PHOTOGEAPHIC  OPTICS 


the  focus  will  be  found  at  a point  nearer  to  the  lens 
than  F. 

Definition, — If  the  incident  rays  be  parallel  to  the 
axis  as  in  the  figure,  then  the  length  A F between  the 
points  in  which  the  central  and  marginal  rays  come  to 
a focus  is  called  the  Longitudinal  Aberration  of  the  lens. 
Also  if  F B be  drawn,  so  that  it  is  the  radius  of  the 
section  of  the  cone  of  rays  by  a plane  at  F perpen- 
dicular to  the  axis,  then  F B is  called  the  Lateral 
Aberration  of  the  lens.  (See  note  on  p.  130.) 

656*.  Calculation  of  Aberration. — We  have  in  the  last 
chapter  considered  the  case  of  rays  passing  symmetrically 
through  the  centre  of  the  lens ; we  have  now  to  take 
the  case  of  rays  passing  through  or  near  the  edge. 

We  shall  consider  the  lens  to  be  thin,  as  the  result 
will  be  quite  close  enough  for  most  purposes.  Let  u be 
the  distance  of  the  object  on  the  axis  from  the  lens,  and 
V the  distance  from  the  lens  at  which  the  marginal  rays 
come  to  a focus,  the  positive  and  negative  directions 
being  taken  as  before.  Let  r and  s be  the  radii  of  the 
front  and  back  surfaces  respectively,/*  the  focal  length 
as  already  found,  jn  the  refractive  index,  and  the  radius 
of  the  aperture  of  the  lens. 

Then  if  2:  is  so  small  compared  with  r and  s that 
we  may  neglect  powers  of  zjf,  zjr,  z/s  beyond  the 
second ; in  place  of  the  old  value  for  v 

uf 

+/ 


1.  e. 


we  now  get  ^ 

uf  __  2 

^ + 7 ^ 

where  A = (2  — ^ 


uf  V 
^ +//  / 


/ t u U^J 


1 


1 


(^  + 2 - 2 — -4- 


rs 


^ See  M.  Martin’s  paper  quoted  above,  or  Coddington’s  Optics. 


ABERRATION 


129 


^ B = (4  + 3 /t  — 3 i + (/t  + 3 /t^)  1 
T s 

fiQ  = 2 + 3 

It  shguld  be  noticed  in  this  relation  that  the  first 
power  of  does  not  occur,  hence  if  the  aperture  is 
such  that  we  can  neglect  squares  of  zjj,  zjr,  z/s,  the 
elementary  relation  will  be  near  enough  to  accuracy 
for  practical  purposes  even  though  we  cannot  neglect 
first  powers  of  these  quantities. 

To  facilitate  calculation  a table  is  given  below,  show- 
ing the  values  of  the  coefficients  in  A,  B,  and  C for 
different  values  of  fx ; the  range  of  jx  is  taken  from  1*50 
to  1*70,  which  includes  all  that  is  usually  needed. 


- - 

1 

i + 3-;v 

1 + 3 fi 

~ + 3 

1-50 

•5834 

— -5000 

1-167 

5-500 

4-333 

1-51 

•5847 

— -5400 

1-119 

5-530 

4-325 

1-52 

•5862 

— -5810 

1-072 

5-560 

4-316 

1-53 

•5881 

— -6220 

1-024 

5-590 

4-307 

1-54 

•5904 

— -6630 

•9776 

5-620 

4-299 

1-55 

•5924 

— -7050 

•9308 

5-650 

4-290 

1*56 

•5956 

— -7470 

•8840 

5-680 

4-282 

1*57 

•5987 

— -7900 

•8376 

5-710 

4-274 

1*58 

•6022 

— -8330 

•7916 

5 740 

4-266 

1-59 

•6059 

— -8760 

•7456 

5-770 

4-258 

1-60 

•6100 

— -9200 

•7000 

5-800 

4-250 

1-61 

•6143 

— -9640 

•6544 

5-830 

4-242 

1*62 

•6190 

— 1-009 

•6092 

5-860 

4-235 

1-63 

•6239 

— 1-054 

•5640 

5-890 

4-227 

1-64 

•6292 

— 1-099 

•5192 

5-920 

4-219 

1-65 

•6347 

— 1145 

•4744 

5-950 

4-212 

1-66 

•6404 

— 1-191 

•4296 

5-980 

4-205 

1-67 

•6465 

— 1-238 

•3852 

6-010 

4-198 

1-68 

•6528 

— 1-285 

•3408 

6-040 

4-190 

1-69 

•6595 

— 1-332 

•2908 

6-070 

4-183 

1-70 

•6664 

— 1-380 

•2528 

6-100 

4-176 

If  the  values  of  the  coefficients  corresponding  to  values 
of  IX  lying  between  those  given  are  required,  they  can 
be  found  by  interpolation  in  the  ordinary  way. 

K 


130 


PHOTOGRAPHIC  OPTICS 


In  photography  the  object  is  often  at  a considerable 
distance,  in  which  case  we  can  get  a close  approxima- 
tion if  we  make  u infinite  or  \\u  zero;  the  expression 


uf 


and  we  ijet 


,,  which  may  be  written 


/ 


1 +fh 


reduces  to  /, 


V =f  - 


f A 


or  the  longitudinal  aberration  AF  (Fig.  47),  usually 
called  a,  is  given  by 

z\f  A 
a = - 

This  is  the  formula  usually  quoted. 

To  find  the  lateral  aberration  denoted  by  6,  we  have 
(Fig.  43)  by  similar  triangles  M L A,  A F B,^ 

F^  _ MJ. 

F A ~ 

Now  = and  we  may  take  L A as  approxi- 

mately equal  to  L F or/*. 


, ML 
or  0 = - — r a 
L A 


'■'’“7“=/ 


f A 


z^  A 


Note. — The  complete  expression  connecting  the  dis- 
tances of  object  and  image  for  a thick  lens  may  be 
interesting  to  some. 

If  e be  the  thickness,  u and  v the  distances  of  the 
object  and  image  from  the  front  and  back  surfaces 
respectively,  the  relation  required  is 


1 1 / 1 

- — h (p—  1)/ 

V u \r 


u / 


+ 


p - 1 /I  1 


B C 


- - A--  + 


where  A,  B,  C have  the  meanings  given  above. 

^ B is  not  shown  in  the  figure ; to  get  it  draw  from  F a per- 
pendicular to  the  axis  of  the  lens  to  cut  the  extreme  ray  on 
either  side  in  B. 


ABERRATION 


131 


The  only  extra  term  is  that  depending  on  e,  terms  in 
e are  omitted,  as  e is  usually  a small  quantity. 

The  other  quantity  required  to  fix  completely  the 
position  of  the  emergent  ray  is  the  angle  which  it  makes 
with  th^  axis  of  the  lens. 

If  6 be  the  angle  made  by  the  incident  ray,  and  r] 
that  made  by  the  emergent  ray  with  the  axis  of  the 
lens,  these  angles  are  connected  by  the  relation 


tan 
tan  £ 


u 

V 


/I  /I  _ 1\ 

'2  fuL  [r  \7"  uj 


If  more  than  one  lens  is  employed  we  can  calculate 
successively  the  angles  which  the  rays  emergent  from 
each  lens  make  with  the  axis. 

66.  Numerical  Examples  of  Aberration. — Consider 
the  convergent  lenses  treated  in  § 33  : — 

(a)  Meniscus  as  in  Fig.  19,  E. 

r = 7 inches,  s = 5 inches,  /x  = 1*5  inch,  and  take 
z = '5  inch. 

. • . A = (2  - 2 . + + (1  + 2^  - 2m2)-1  + 

\j.L  jr^  r s 


_ *5834  *5000  , 2*25 
“ ~49  ^ 

= -0876. 


•0119  - -0143  + -0900 


From  § 33  we  gety  the  focal  length  = 35  inches. 


^/■A 


•25  X 35  X -0876 


= *383  inch. 


If  the  lens  be  reversed  we  get 
r = — 5 inches,  s = — 7 inches. 


A = 


*5834 

“25~^ 


5000  2*25 

35  “49~ 


*0232  - *0143  + 


*0459  = *0458 


*25  X 35  X *0548 


= *201  inch. 


2 


132 


PHOTOGRAPHIC  OPTICS 


(h)  Double  convex  lens  as  in  Fig.  19,  A. 
r = — 7 inches,  s = 5 inches,  //  = 1*5,^  = — 5*83 


inches. 
Hence  A = 


5834  -5000 

49  35 


2-25 

= -0119  + -0143 
25 


. a 


+ -0900  = -1162 

z^fA  -25  X 5-83  X *1162  , 

= = *085  inch. 


If  the  lens  be  reversed  we  get 
r = — 5 inches,  5=7  inches. 


;^34  -50^  2;_25  _ 

25~  35  'W 


•0232  + -0143  + 
•0459  = -0834 


f A 

a = — V — ^ 


•25  X 5-83  X -0834 

2 


= •OGl  inch. 


In  these  cases,  the  lenses  being  totally  uncorrected, 
the  aberration  is  larger  than  could  be  tolerated  in 
practice,  but  the  results  serve  to  illustrate  one  or  two 
important  points. 

In  the  first  part  of  example  (a)  when  the  meniscus 
lens  is  placed  with  its  concave  face  towards  the  incident 
pencil  of  parallel  rays,  the  aberration  is  nearly  double 
what  it  is  when  the  lens  is  reversed.  Examination  of 
diagrams  will  show  that  the  angles  of  incidence  and 
refraction  at  both  surfaces  are  less  in  the  latter  than  in 
the  former  case. 

This  is  an  example  of  the  general  principle  that  the 
aberration  of  a lens  depends  on  the  nature  of  the  face 
which  receives  the  incident  light,  but  that  the  smaller 
the  angle  of  incidence  and  the  smaller  the  angle  of 
refraction  the  less  is  the  aberration.  To  secure  this 
condition  with  a single  lens  or  cemented  combination, 
for  incident  rays  parallel  to  the  axis,  the  face  whose 
radius  of  curvature  is  greatest,  whether  convex  or 
concave,  must  receive  the  incident  light,  and  this  is 
usually  the  case  with  simple  objectives, 


ABERRATION 


133 


For  compound  objectives  the  problem  is  not  quite  so 
simple,  as  the  rays,  after  passing  through  the  first 
combination,  are  not  parallel  for  incidence  as  the  second 
combination. 

It  will  be  found  in  the  majority  of  cases,  that  in  each 
combination  the  radii  of  curvature  of  the  outside 
surfaces  are  greater  than  those  of  the  faces  cemented 
together,  and  as  a rule  the  flattest  faces  of  the 
combinations  face  each  other. 

67.  Aberration  for  two  Lenses. — When  there  are 
two  lenses  in  contact,  let  F be  the  focal  length  of  the 
combination,  / and  /'  those  of  the  component  lenses,  and 
let  the  quantities  for  the  second  lens  corresponding 
to  those  for  the  first  lens  be  denoted  by  dashed  letters  ; 
then  the  value  of  v for  the  marginal  rays  is  given  by 


where  A,  B,  C,  etc.  have  the  values  assigned  to  them  in 
I 65.  If  the  object  is  distant,  and  hence  very  large, 
we  get  for  the  longitudinal  aberration — 

(A  , A'  B'  , C' 

17  + 7 -J'+77 

For  the  treatment  of  three  or  more  lenses  in  contact, 
reference  should  be  made  to  M.  Martin’s  paper. 

68.  Trigonometrical  Method. — The  method  already 
given  is  useful  for  designing  lenses,  but  when  the 
aberration  of  a given  lens  is  required  a more  direct 
method  may  be  adopted. 

The  method  is  that  of  tracing  the  course  of  a ray, 
originally  parallel  to  the  axis  of  the  lens  at  any  required 
distance  till  it  cuts  the  axis ; this  will  give  the  position 
of  the  point  A (Fig.  47),  the  position  of  F can  be  found 
in  the  usual  way,  and  thus  A F,  the  aberration,  will  be 
known. 

Use  the  following  notation  in  the  calculations  : 


134 


PHOTOGRAPHIC  OPTICS 


etc.,  are  the  refractive  inches  of  the  lenses ; R2, 

Rg,  etc.,  are  the  radii  of  the  successive  surfaces  ; a is 
the  angle  of  incidence,  and  a the  corresponding  angle 
of  refraction  at  the  first  surface,  b and  /5  the  corre- 
sponding angles  for  the  second  surface,  and  so  on.  The 
elongations  of  the  successive  portions  of  the  ray,  after 
the  refractions  which  it  undergoes,  cut  the  axis  in 
points  called  A,  B,  C,  etc.  ; the  distances  of  these  points 
from  the  first,  second,  third,  etc.  surfaces  respectively 
are  called  A,  B,  C,  etc.,  and  the  angles  which  they 
make  with  the  axis  (A),  (B),  (C)  etc.,  and  e^  e2,  etc.  are 
the  distances  between  the  successive  surfaces  measured 
along  the  axis. 

The  different  portions  of  the  refracted  ray  make 
with  the  axis  and  the  radii  of  the  surfaces  a series  of 
rectilineal  triangles  which  can  be  solved  in  succession. 

The  formulae  required  should  in  each  case  be  written 
down  from  a consideration  of  the  particular  figure. 

In  the  case  of  a double  convex  front  lens  in  contact 
with  a double  concave  lens,  the  two  being  cemented,  the 
following  will  be  the  for  mu  he  required ; they  should  be 
verified  from  a figure. 

The  incident  ray  is  parallel  to  the  axis  and  at  a 
distance  2;  from  it  : — 

For  the  first  refraction — 


, Sin  a 

sin  a = — , sin  a = 

Ri 


^1 


I / A \ A Sin  CL  -pj 

(A  =a-o,A  = 

y Sin  (A) 

For  the  refraction  from  the  first  to  the  second  lens — 
sin  h = ^ sin  (A), sin  /3  = — sinh,  ^ ^ ) 


II 


A + Ro  ■—  61  . 

^ i sir, 

1^2 


For  the  third  refraction,  out  at  the  last  surface — 


ABERRATION 


135 


R. 


B + ^2 


III 


Rq 


, sin  y = 1^2  ^ 


(C)  = (B)  + c - y , C = + Ks 

' Sin  (C) 


As  a numerical  example,  consider  the  lens  in  § 33. 
r = 7 inches,  s = 5 inches,  /m  = 1 ’5,  and  let  z = 'b  inch. 

For  the  first  surface  the  formulse  (I)  above  are 
applicable,  the  results  of  the  calculation  are — 

a = 4 o 46  I ^ ^ 13-976  + 7 
= 20*976  inches 


ri  = 2°  43'  46 
(A)  = 1°  22'  0 

For  the  second  surface  the  formulae  are — 


■j 


din  h = ^ sin  (A)  , sin  j3  — fi  sin  h , 


n. 


(B)  = /3-A-6,B- 


The  results  are — 

b = 4°  22M4"'l  ^ , 

y/  I B = 


Rj  sin  13 
sin  (B) 


- R2 


13  = 6°  33'  50' 
(B)  = 0°  49'  36' 


39-615 
j = 34-615 


Hence  a ray  which  before  incidence  is  parallel  to  the 
axis  and  half  an  inch  from  it,  after  refraction  cuts  the 
axis  at  a distance  of  34*615  inches  from  the  lens;  now 
the  focal  length  of  the  lens  (§  33)  is  35  inches,  hence 
we  have  for  the  aberration 

a = 35*000  - 34*615  = *385  inch. 

69.  Minimum  Aberration. — We  have  seen  (? 
that  the  lateral  aberration  of  a lens  is  given  by 


65) 


'■/ 


where 


, A = 


2 - 2 fx‘^  + 


r s s'^ 


If  the  aberration  is  to  vanish  (for  terms  as  far  as  z^) 


136 


PHOTOGRAPHIC  OPTICS 


we  must  have  A = 0.  This  gives  us  a quadratic 
equation  to  determine  the  ratio  of  r : s when  the 
refractive  index  fj,  is  given ; it  can  be  shown  that  if 
the  roots  of  this  equation  are  to  be  real,  we  must  have 
fjb  less  than  one  quarter ; no  substance  is  known  which 
has  such  a refractive  index,  and  hence  a single  lens  free 
from  aberration  cannot  be  made. 

We  can,  however,  choose  the  ratio  r : s when  the 
value  of  jLi  is  given  to  make  the  aberration  a minimum, 
and  if  we  wish  the  lens  to  be  of  given  focal  length,  we 
can  completely  determine  r and  s. 

The  two  equations  for  finding  r and  s are  (Wallou, 
p.  275)— 

r 4 + — 2 1 


/ 


= (/^ 


1)  (I -I 

' r s 


where  f is  the  focal  length  required. 

Such  a lens  is  called  a crossed  lens. 

Example. — The  following  is  taken  from  Wallon. 
Let  p = 1*5;  then 


4 + /X  — 2 


s fjb  2 fjir  6 

The  negative  sign  shows  that  the  radii  of  curvature  of  the 
surface  must  be  in  opposite  directions,  hence  the  lens 
must  either  be  double  convex  or  double  concave. 


We  have  also  y =(^  — 1) 


1 


= -5  ( i - 1 

Si  \r  s 


whence  we  get — 


^ 12-^’ 


If  the  lens  is  convergenty*is  negative,  which  makes  r 
negative  and  s positive,  or  the  lens  is  double  convex,  as 
we  should  expect. 

If  these  values  of  r and  s are  used  to  calculate  the 
aberration  we  get — 


ABERRATION 


137 


a 


14  / 


If  the  lens  be  turned  round  so  that 


we  shall  get  for  the  aberration 


/ 

a 


45  y 

r4  7’ 


which  is  3 a. 


or  the  aberration  in  the  latter  case  is  three  times  as 
large  as  in  the  former. 

The  form  of  the  lens  of  least  aberration  changes 
with  the  refractive  index  of  the  substance  employed  ; 
looking  at  the  expression  for  rjs  we  see  that  the  ratio 
will  be  negative  only  so  long  as 


or 


4+/A  — 

^ — 4<0 


which  is  the  case  only  so  long  as 

1*686 

If  /X  = 1*686,  then  2 — fi  — 4:  — o 

or  the  ratio  rjs  — 0,  and  as  r cannot  be  zero  - must  be 

zero,  or  the  back  face  of  the  lens  is  in  this  case 
plane. 

70.  Oblique  Pencils. — Now  take  the  case  of  a 
pencil  of  rays  striking  the  lens  obliquely.  If  the 
region  behind  the  lens  be  explored  by  means  of  a 
screen  (as  in  | 64)  the  appearances  on  it  will  be  like 
those  in  Fig.  47a,  which  is  reproduced  from  photographs 
of  the  actual  phenomena.  The  arrangements  in  this  case 
are  similar  to  those  in  § 64,  with  the  exception  that 
the  lens  is  turned,  round  a vertical  axis  through  its 
centre  through  a suitable  angle.  A cursory  examina- 
tion of  the  figures  shows  that  the  rays,  after  refraction, 


138 


PHOTOGKAPHIC  OPTICS 


do  not  pass  nearly  through  one  point,  but  are  in  a 
seemingly  inextricable  jumble. 


Fig.  ila  (1), 


Fig.  47«  (2). 


If,  however,  a diaphragm  is  placed  in  front  of  the 
lens,  so  that  the  central  portion  only  is  used,  the  figures 
corresponding  to  the  former  become  now  those  of  Fig. 


ABERRATION 


139 


476;  here,  if  the  screen  be  placed  near  the  lens,  the 
appearance  is  an  elliptical-shaped  figure;  on  moving  the 


Fig.  47a  (3). 


Fig.  47a  (4). 

screen  away  from  the  lens  the  ellipse  shrinks  very 
nearly  into  a straight  line,  then  it  broadens  out  and 
approximates  to  a circle.  On  further  movement  of  the 


140 


PHOTOGRAPHIC  OPTICS 


screen  this  circle  lengthens  out  into  a straight  line  at 
right  angles  to  the  former,  and  this  again  broadens  out 
into  an  oval. 


Fig.  m (5). 


Fig.  4:7a  (6). 

The  two  lines  thus  found  are  called  focal  lines,  and 
play  an  important  part  in  the  theory  of  oblique 
pencils. 


ABERRATION 


141 


We  thus  arrive  at  an  important  result,  which  can  be 
shown  to  hold  generally  : — 


Fig.  m (1). 


Fig.  m (2). 


Tf  a small  pencil  proceeding  from  a luminous  point 
pass  obliquely  through  any  refracting  surfaces, 
the  rays  after  any  number  of  refractions  pass 


142 


PHOTOGRAPHIC  OPTICS 


approximately  through  two  straight  lines  at 
right  angles. 


^Fig.  m (3). 


Fig.  47&  (4). 


It  is  not  altogether  easy  to  form  a clear  idea  of  the 
shape  of  the  pencil  after  refraction,  but  the  imagination 
may  be  aided  by  a model  which  is  not  hard  to  con- 


ABERRATION 


143 


struct.  Take  a cardboard  box,  and  on  opposite  sides 
of  it  mark  out  two  lines,  at  right  angles  ; in  these  lines 


Fig.  47&  (5). 


Fig.  47&  (6). 

pierce  holes  at  convenient  distances  apart  (five  in  each 
will  be  enough),  and  pass  a thread  from  every  hole  of 
one  set  to  every  hole  of  the  other  ; if  the  sides  of  the 


144 


PHOTOGRAPHIC  OPTICS 


box  are  now  pulled  apart  until  the  threads  are  taut, 
we  shall  have  a fair  model  of  the  pencil. 

The  circular-shaped  patch  of  light  at  a point  be- 
tween the  lines  is  called  the  circle  of  least  confusion, 

71.  Let  us  now  examine  how  such  an  arrange- 
ment  of  the  rays  originates.  Since  the  characteristics 
of  all  small  pencils  of  rays  proceeding  from  a point,  or 
parallel,  are,  after  refraction,  the  same,  let  us  take  a 
case  which  is  rather  more  simple  than  an  oblique 
central  pencil,  though  not  so  easy  to  realize  experi- 
mentally. 

Consider  a small  pencil  of  rays  parallel  to  the  axis, 
but  striking  the  lens  at  some  distance  from  the  centre 
of  the  lens ; let  the  lens  be  placed  with  its  axis  hori- 
zontal, with  the  pencil  vertically  above  the  axis. 

Looked  at  from  the  side  the  pencil  would  present 
the  appearance  of  Fig.  47c,  which  may  be  obtained 
from  Fig.  47  by  erasing  all  the  rays  not  required.  The 
rays  retained  form  a small  part  of  the  caustic  surface 
near  H,  and  those  in  the  plane  of  the  paper  pass  very 
nearly  all  through  that  point.  It  therefore  at  first 
sight  looks  as  if  H is  the  image,  or  the  focus  of  the 
pencil ; but  this  is  not  the  case. 

The  pencil  if  seen  from  above  would  appear  as  in 
Fig.  47(i,  in  which  the  letters  correspond  with  those 
of  Fig.  47c. 

We  have  to  see  how  the  focal  lines  arise ; in  Fig. 
47c  we  have  a section  only  of  the  pencil;  the  pencil 
itself  has  of  course  sensible  thickness.  We  may 
imagine  the  pencil  itself  to  be  produced  by  rotating 
Fig.  47c  through  a small  angle  about  the  axis  of  the 
lens ; H will  remain  always  at  the  same  distance  from 
the  axis,  and  therefore  will  trace  out  the  arc  of  a circle, 
which,  being  short,  may  be  regarded  as  a straight 
line. 

Hence  all  the  rays  of  the  pencil  pass  approximately 
through  this  line,  which  is  therefore  a focal  line.  On 
the  other  hand,  let  the  central  ray  of  the  pencil  cut 


ABERRATION 


145 


L 


146 


PHOTOGRAPHIC  OPTICS 


ABERRATION 


147 


the  axis  in  A,  and  draw  NAM  perpendicular  to  the 
general  direction  of  the  rays  ; the  revolution  will  turn 
this  line  through  a small  angle,  and  it  will  trace  out  a 
figure  like  that  in  Fig.  48,  a sort  of  rough  figure  of  8, 
through  which  all  the  rays  pass ; being  slender  this 
may  roughly  be  taken  as  a line,  and  is  the  other  focal 
line.  Thus  all  the  ra3^s  of  the  pencil  pass  very  nearly 
through  two  straight  lines  at  right  angles. 

As  we  pass  along  the  pencil  from  H to  A it  will 
contract  horizontally  and  extend  vertically  so  that  at 
some  intermediate  point  the  pencil  is  of  the  same 
breadth  in  both  directions  and  is  roughly  a circle,  the 
circle  of  least  confusion. 

A similar  course  of  reasoning  will  show  the  existence 
of  focal  lines  in  the  case  of  a small  oblique  pencil 
striking  the  lens  at  its  centre. 

72.  General  Theory  of  Focal  Lines. — 

The  general  theory  rests  on  the  following 
two  proportions  : — 

(a)  If  a pencil,  to  begin  with,  is  such 

that  its  rays  are  all  normal  to  some  Fig.  48. 
surface,  then  after  any  number  of 
reflections  and  refractions,  there  will  still  be 
some  surface  to  which  they  are  normal. 

A simple  example  of  this  is  a pencil  of  rays  from  a 
point  reflected  at  a plane  surface  ; before  reflection  the 
rays  are  all  normals  to  spheres  whose  centre  is  the 
source  of  light,  and  after  reflection  they  are  normal  to 
spheres  whose  centre  is  the  image  of  the  source. 

This  is  shown  in  Fig.  49,  where  the  dotted  circles 
are  sections  of  the  spheres;  here  the  effect  of  the 
reflection  is  simply  to  reverse  the  surface  without 
changing  its  nature,  but  in  most  cases  a reflection  or 
refraction  will  totally  change  the  nature  of  the 
surface. 

The  surface  in  question  is  that  which  in  Physical 
Optics  is  called  the  Wave  Surface,  and  is  in  fact  the 
shape  of  the  wave  starting  from  the  given  source,  after 


148 


PHOTOGRAPHIC  OPTICS 


a certain  time ; the  effect  of  reflection  or  refraction 
being  to  bend  the  wave  into  various  shapes. 


ABEREATION 


149 


The  well-known  effect  of  throwing  a stone  into  a 
still  pond  in  causing  circular  waves  to  travel  outwards 
is  an  example  of  this. 


150 


PHOTOGRAPHIC  OPTICS 


The  action  of  a lens  in  bringing  rays  proceeding 
from  one  point,  to  a focus  at  another,  may  be  explained 
by  saying  that  the  lens  twists  the  wave  surface  from 
one  sphere  to  another,  as  in  Fig.  50  ; which  is  due  to 
the  fact  that  light  does  not  travel  as  fast  in  glass  as  in 
air.  The  central  portions  of  the  wave  have  to  travel 
through  a greater  or  less  thickness  of  glass  (according 
to  the  nature  of  the  lens)  than  the  extreme  portions, 
and  thus  are  either  overtaken  by  or  overtake  the  extreme 
portions,  from  which  the  change  in  the  shape  results. 

The  second  proposition,  which  is  given  in  books  as 
Analytical  Solid  Geometry,^  is — 

(6)  The  lines  normal  to  a small  area  of  any  surface 
(provided  there  is  no  edge  or  point  in  the  area) 
pass  approximately  through  two  straight  lines 
at  right  angles. 

From  {a)  we  learn  that  the  rays  in  the  pencils  with 
which  we  have  to  do,  since  they  are,  to  begin  with, 
normal  either  to  a sphere  or  a plane  (because  they 
either  proceed  from  a point  or  are  parallel),  are  always 
normal  to  a surface,  called  the  wave  surface. 

And  since  they  are  normals  to  a small  portion  of  the 
wave  surface,  we  learn  from  (h)  that  they  pass"approxi- 
mately  through  two  straight  lines  at  right  angles. 

This  proves  the  statement,  made  above,  that  all 
small  pencils  which  are  oblique  to  refractory  surfaces 
have  focal  lines  after  refraction. 

When  the  pencil  strikes  the  surfaces  symmetrically 
the  focal  lines  coincide  and  reduce  either  to  a small 
circle  or  point,  as  already  described  in  §-64. 

73.  Central  Oblique  Pencil. — Consider  now  a small 
pencil  striking  the  lens  obliquely  at  its  centre ; in  this 
case  we  can  give  the  expressions  for  the  distances  of 
the  focal  lines  from  the  lens. 

Take  the  first  refraction  of  the  pencil  through  one 
spherical  surface;  this  is  reproduced  in  Fig.  51;  in 
which  the  size  of  the  pencil  is  purposely  very  much 
1 Frost’s  Solid  Geometry,  1875,  p.  388,  § 588, 


ABERRATION 


151 


exaggerated  to  avoid  confusion  ; the  pencil  is  supposed 
to  be  divided  up  by  vertical  and  horizontal  planes. 


152 


PHOTOGRAPHIC  OPTICS 


The  rays  in  the  vertical  planes  P A2  A3,  P 
B2  B3,  P C2  C3  converge  after  the  refraction  to  points 
a,  6,  c,  respectively,  which  form  the  horizontal  focal 
line  ; and  the  rays  in  the  planes  P A;^  B^^  C^,  P A2  B2  C2, 
P Ag  Bg  C3,  at  right  angles  to  the  former,  to  D,  E,  F 
respectively,  which  form  the  other  focal  line. 

Let  cfi  be  the  angle  which  the  axis  P B of  the  pencil 
makes  with  the  axis  of  the  lens,  and  the  corresponding 
angle  after  refraction ; let  be  the  distances  of  h 

and  E from  B (the  positive  and  negative  directions 
being  reckoned  as  before).  Then  it  can  be  proved  that 
the  relations  between  are  ^ — 

/X  co^  (f)'  cos^  0 fx  cos  cp'  — cos  (p 
Wi  V r 

fJL  1 fX  COS  (j/  — cos  0 

W2  V r 

Both  these  relations  reduce  to  that  already  found  for 
a spherical  surface  if  we  put  ^ = 0,  = 0,  and  are  in 

fact  an  extension  of  the  previous  formula.  Now  let 
the  pencil  strike  a second  surface  of  radius  s,  and  let 
and  V2  be  the  distances  of  the  focal  lines  from  the 
surface,  the  lens  being  taken  as  thin. 

At  this  second  refraction,  the  horizontal  focal  line 
of  the  first  refraction,  being  due  to  rays  in  vertical 
planes,  will  evidently  give  rise  to  the  horizontal  focal 
line  after  the  second  refraction,  and  similarly  the 
remaining  focal  lines  in  the  two  cases  will  correspond. 

The  axis  of  the  pencil,  which  passes  undeviated,  will 
after  refraction  at  the  second  surface  be  inclined  to 
the  axis  at  an  angle  </>,  and  the  refractive  index  for  this 
second  refraction  is  l/fx;  hence  the  formulae  for  the 
second  refraction  may  be  got  from  those  for  the  first 
refraction  by  interchanging  </>,  and  cp'  and  putting  l/fx 
for  IX,  we  thus  get 

^ Aldis,  Geometrical  Optics,  ed.  3,  Arts.  46,  73. 


ABERRATION 


153 


— COS^  0 


^1 

1 


„ — cos  0 — COS  0' 

COS^  0 _ /X 

Wi  s 

— cos  0 ~ COS  0' 

1 


'^2  '2^2  ^ 

or  multiplying  throughout  by 

0 /X  ^ 

11  /Li  cos  0'  — cos  0 


'w;.? 


Adding  these  to  the  former  equations  we  get  for  the 
relations  between  t?.,  Vo  and  u for  a thin  lens — 


(/Li  cos  (j)'  — cos  0)( 


cos^  (p  cos"^  0 


= (jLL  cos  d)'  — cos  (h)  ( 

v^u  ^ \r  s 

These  relations  reduce  to  those  already  found  for 
a thin  lens  if  9 and  (p'  vanish. 

When  the  incident  pencil  consists  of  parallel  rays  we 
must  put  1/^^  = 0,  and  we  get 


cos^  0 


1 


V-^  V 

which  shows  us  that 
greater  than  for 


= - = (^  cos  (j)'  — cos  (f)  I 


1 1 


can  never  be  numerically 


COS^  (f) 


and  the  cosine  of  an  angle  cannot  be  greater  than 
unity.  Hence  the  focal  line  which  is  perpendicular  to 
the  plane  containing  the  axes  of  both  the  pencil  and 
the  lens  is  nearer  to  the  lens  than  the  other  focal  line. 

73a.  Construction  for  Focal  Lines. — When  the  pencil 
incident  on  a single  spherical  surface  is  composed 
of  parallel  rays  we  may  find  the  position  of  the  focal 
lines  as  follows — ^ 


154 


PHOTOGRAPHIC  OPTICS 


Let  P Q be  the  mean  ray  of  the  pencil  (Fig.  51a)  and 
R S a near  ray ; through  the  centre  O of  the  surface 


ABERRATION 


155 


draw  O A parallel  to  the  incident  pencil,  and  let  it 
meet  P Q,  R S,  after  refraction,  in  C and  D,  and  let  Q C, 
S D intersect  in  X. 

If  now  the  figure  be  imagined  to  be  rotated  through 
a small  angle  round  O A,  X will  trace  out  the  focal 
line  perpendicular  to  the  paper  ; the  other  focal  line 
will  be  at  C,  for  all  the  rays  intersect  O A near  this 
point. 

Thus  to  get  the  position  of  the  focal  line  lying  in  the 
plane  of  the  paper  we  must  draw  a radius  parallel  to 
the  incident  pencil  and  find  the  point  where  this  is  cut 
by  the  mean  ray  after  refraction. 

74.  Distance  between  Focal  Lines  after  Refraction  at 
the  First  Surface. — We  shall  consider  in  this  article  the 
case  only  of  pencils  with  parallel  rays  ; here  we  must 
put  l/u  = 0,  and  the  distances  of  the  focal  lines  from 
the  surface  are  given  by — 

/t  (}>'  jUL  fjL  cos  — cos  0 
Wo  r 

Now  sin  0 = /M,  sin  (j)\  and  therefore  = sin  0 / sin  <p' 


, sin  0 , 

. * . /A  COS  0 --  cos  0 = -T r COS  0 — COS  0 = 

Sin  0 

sin  0 cos  (p'  — cos  0 sin  (p'  sin  (0  — (p') 
sin  0'  sin  0' 

Substituting  this  we  get  easily 

ytt  0'  sin  0'  fii  sin  (b' 

Sin  {(p  — (p  ) sin  {(p  --  (p) 


jLL  sin  0(1—  cos^  0 ) fj  sin^  0 

. *.  Wo  — w.  = r ^ K---  = r -f— 

sin  (0  — 0 ) sin  (0  — 0 ) 

If  the  angle  of  incidence  is  small  we  can  replace  sin  0 
and  sin  (0  — 0')  by  the  circular  measure  of  the  angles, 
and  we  get 

Wt  = r , — — - and  this  becomes 


^ ^ (9-  ¥)  ^ (m  - 1) 

for  the  relation  sin  <[)  = fi  sin  (f>'  reduced  to  0 = /<  0'. 


156 


PHOTOGRAPHIC  OPTICS 


This  shows  us  that  the  distance  w<2^  — between 
the  focal  lines,  increases  with  the  inclination  (p  of  the 
axis  of  the  incident  pencil  to  the  axis  of  the  lens,  if  <j) 
is  small ; this  can  be  proved  to  be  true  also  when  the 
inclination  is  considerable. 

A separation  of  the  focal  lines  after  the  first 
refraction  will  obviously  tend  to  produce  a separation 
of  the  focal  lines  after  the  second  refraction  ; we  see 
therefore  that  with  a lens  the  greater  the  obliquity  the 
greater  is  the  distance  between  the  focal  lines. 

74a.  Effect  of  Spherical  Aberration  on  the 
Picture. — It  is  hardly  necessary  to  point  out  that  the 
spherical  aberration  of  a lens  tends  to  destroy  the 
sharpness  and  clearness  of  the  picture. 

Each  luminous  point  of  the  object,  near  the  axis  of 
the  lens,  instead  of  being  represented  in  the  picture  by 
a luminous  point,  as  by  the  elementary  theory  it  should 
be,  is  represented  by  a patch  of  light  which  no  amount 
of  focussing  can  reduce  in  size  beyond  a certain  limit. 
If  these  patches  are  small  enough  not  to  subtend  an 
angle  of  more  than  one  minute  (§  90)  when  viewing 
the  picture  at  a convenient  distance,  the  effect  produced 
is  the  same  as  if  they  were  points ; but  if  they  are 
larger  than  this  they  have  a visible  size  and  overlap, 
thus  confusing  the  images  of  points  near  together  and 
causing  indistinctness. 

In  Fig.  42,  C D is  the  section  of  least  area,  and  it  can 
be  shown  that  the  distance  of  C D from  F is  three- 
fourths  of  A F,  and  the  diameter  of  the  circle  at  C D is 
one-half  the  lateral  aberration  F B.^ 

75.  Astigmatism.  The  effect  of  aberration  on 
pencils  which  pass  obliquely  is  called  astigmatism. 

We  have  seen  that  when  the  pencil  is  oblique  and 
the  aperture  of  the  lens  is  large  its  section  is  nowhere 
even  approximately  a point,  but  is  a considerable  patch 
of  light  varying  in  size  at  different  places. 

But  if  the  pencil  be  restricted  by  a diaphragm  to  be 
^ Coddington’s  Optics,  pp.  11,  12, 


ABERRATION 


157 


fairly  small,  then  the  rays  pass,  not  through  a point 
but  very  approximately  through  the  straight  lines  at 
right  angles,  and  the  nearest  approach  to  a focus  is  at 
some  point  between  them,  where  the  section  of  the 
pencil  is  nearly  circular,  called  the  circle  of  least  confusion. 
If  the  aperture  used  is  small,  this  circle  may  be  made 
small  enough  to  be  practically  a point,  which  can  be 
regarded  as  the  focus. 

If  such  foci  were  all  in  one  plane  perpendicular  to 
the  axis  of  the  lens  and  the  plate  were  placed  in  this 
plane  the  picture  would  be  in  focus  all  over ; but  in 
most  cases  the  circles  of  least  confusion  lie  on  a curved 
surface,  passing  through  the  principal  focus  and  usually 
concave  towards  the  lens,  producing  curvature  of  the 
field. 

When  this  is  the  case  there  is  no  position  in  which 
the  plate  can  be  placed  that  the  foci  of  all  the  pencils 
may  lie  on  it ; if  it  be  placed  to  bring  the  central 
portion  of  the  picture  into  focus,  it  will  cut  very 
oblique  pencils  either  at  or  near  a focal  line,  giving  rise 
to  a line  or  elongated  patch  of  light. 

Let  us  consider  the  nature  of  the  picture  formed  by 
such  lines  or  elongated  patches ; let  the  object  be  a 
cross  (Fig.  52,  1).  If  the  cross  be  placed  at  right  angles 
to  the  focal  line  on  the  screen,  every  point  on  the  vertical 
line  will  be  broadened  out,  each  point  of  the  object 
giving  rise  to  a line  or  elongated  patch ; the  effect  of 
this  is  to  represent  the  cross  by  an  indistinct  broad 
blur  (Fig.  52,  2),  the  only  distinct  portion  being  the 
crossbar,  for  here  the  patches  of  light  overlap  and 
lengthen,  but  do  not  broaden  it  j if  the  cross  be  parallel 
to  the  focal  line  it  is  not  hard  to  see  that  the  general 
form  of  the  image  will  be  that  in  Fig.  52  (3),  and  if 
the  screen  be  placed  to  receive  the  circles  of  least 
confusion,  then  the  effect  is  shown  in  Fig.  52  (4),  in 
which  the  size  of  the  circle  is  exaggerated  to  show  the 
nature  of  the  phenomenon  clearly. 

Careful  inspection  of  these  figures  will  show  that  the 


168 


PHOTOGRAPHIC  OPTICS 


general  effect  of  using  the  section  of  oblique  pencils  near 
focal  lines  is  to  produce  fairly  sharp  images  of  straight 
lines  parallel  to  the  focal  line  used,  but  only  diffuse 
images  of  lines  at  right  angles  to  these,  so  that 
practically  a picture  is  produced  only  of  lines  parallel 
to  the  focal  line  used. 


1 


Fig.  52. 


This  effect  of  producing  an  image  of  lines  in  one 
direction  but  not  of  those  at  right  angles  is  astigmatism; 
it  occurs  not  only  in  photographic  lenses  but  is  a not 
uncommon  defect  of  sight  produced  by  some  deformation 
of  the  lens  of  the  eye. 

In  photographic  lenses  since  the  surface  containing 
the  circles  of  least  confusion  bends  towards  the  lens, 
it  is  usually  the  focal  line  furthest  away  from  the  lens 
which  falls  on  the  plate;  this  line  is  that  one  which  is 


ABEREATION 


159 


radial  or  points  towards  the  axis  of  the  lens,  hence  if 
the  astigmatism  is  sensible  the  images  of  lines  pointing 
towards  the  centre  of  a picture  will  be  sharper  than 
those  of  lines  at  right  angles  to  them ; but  in  most 
cases  the  phenomenon  will  not  be  clearly  marked,  a 
general  indistinctness  at  the  edges  of  the  picture  being 
mainly  observable. 

The  distance  between  the  focal  lines  of  the  pencil  is 
generally  taken  to  be  the  measure  of  its  astigmatism. 

76.  Experiment  on  Astigmatism. — The  remarks  of 
the  preceding  section  can  be  illustrated  by  a simple 
experiment.^ 

Place  upright  in  front  of  a lamp  or  gas  flame  a piece 
of  wire  gauze,  the  wires  being  horizontal  and  vertical, 
and  behind  this  a single  lens  (fitted  with  a diaphragm 
if  necessary),  not  parallel  to  the  gauze  but  twisted 
about  a vertical  axis  through  an  angle  of  about  45° ; if 
the  image  be  received  on  a vertical  screen  it  will  be 
found  that  the  horizontal  wires  are  in  focus  in  one 
position  of  the  screen  and  the  vertical  wires  in  another. 

If  the  lens  be  at  a considerable  distance  from  the 
gauze  so  that  Iju  can  be  neglected,  and  be  the 

distances  of  the  focal  lines  from  the  lens,  we  have  (§73) 

cos^  (i  = — 

^2 

hence  if  the  inclination  of  the  lens  to  the  direction  of 
the  light  can  be  measured  we  can  roughly  verify  the 
formulse  given. 

77.  Distortion  due  to  the  Use  of  a Diaphragm. — A 

diaphragm  is  used  with  a lens  to  reduce  the  size  of 
the  pencil,  thus  diminishing  the  size  of  the  circle  of 
least  confusion,  and  improving  the  definition. 

The  best  effect  is  produced  when  the  diaphragm  is 
not  in  contact  with  the  lens,  but  at  a short  distance 
from  it,  so  that  the  pencils  strike  the  lens  less  obliquely, 
and  there  is  less  astigmatism ; in  fact  the  arrangement 

1 Glazebrook  and  Shaw’s  Practical  Physics,  4th  Ed.  p.  354. 


160 


PHOTOGRAPHIC  OPTICS 


combines  the  advantages  of  the  use  of  a small  pencil 
with  incidence  as  nearly  direct  as  possible. 

But  this  arrangement,  though  it  produces  a beneficial 
effect  on  the  sharpness  of  the  picture,  introduces 
another  defect  which  would  be  noticeable  when  the 
lens  is  used  for  copying  or  architectural  work ; a 
distortion  is  produced,  the  nature  of  which  we  proceed 
to  examine. 

This  distortion  must  be  clearly  distinguished  from 
that  which  we  have  already  considered,  which  was  due 
to  the  wrong  position  of  the  plate  relative  to  the 
object,  causing  lines  which  should  be  parallel  to  run 
together,  which  can  be  corrected  by  the  use  of  the 
swing  back. 

Here  the  effect  is  not  to  make  lines  converge,  but  to 
bend  them,  making  lines  near  the  edge  of  a picture 
curved  instead  of  straight.  The  phenomenon  is  best 
illustrated  experimentally,  but  as  the  effect  is  usually 
small  it  must  be  looked  for  carefully,  and  the  straight 
lines  on  the  picture  tested  with  a straight  edge. 

In  Fig.  53  is  given  a horizontal  section  of  the 
arrangement,  through  the  centre  of  the  lens.  In  front 
of  a screen  to  receive  the  image  the  lens  is  placed 
upright,  both  being  movable ; in  front  of  the  lens  is 
placed  a movable  diaphragm — a piece  of  cardboard 
pierced  with  a small  clean  circular  hole  will  do — the 
hole  being  on  the  axis  of  the  lens ; to  one  side  a 
straight-edged  rule  is  placed  vertically,  and  behind  it 
a lamp  with  a large  plain  globe.  If  a lamp  with  a 
globe  is  not  available  a piece  of  ground  glass  may  be 
used  instead  of  the  globe  to  make  a uniform  back- 
ground to  a considerable  length  of  the  vertical  rule. 

The  diagram  is  not  drawn  to  scale,  the  lamp  and 
rule  being  considerably  nearer  to  the  lens  than  they 
should  be  in  the  actual  experiment. 

If  the  diaphragm  is  near  the  lens  the  image  of  the 
straight  edge,  formed  on  the  screen,  is  straight,  but  if 
the  diaphragm  be  moved  backwards  the  image  will 


ABERRATION 


161 


become  curved,  the  line  being  concave  towards  the 
axis  of  the  lens. 


In  the  diagram  the  pencil  coming  from  A the  edge 
of  the  rule  is  traced,  it  falls  on  the  edge  of  the  lens, 

M 


162 


PHOTOGRAPHIC  OPTICS 


and  it  is  to  this  that  the  phenomenon  is  to  a large 
extent  due. 

It  will  be  noticed  that  the  curvature  of  the  image 
increases  when  the  diaphragm  recedes  from  the  lens 
and  decreases  when  it  approaches  it. 

It  should  be  noticed  that  as  the  diaphragm  recedes 


1 


the  pencil  strikes  the  lens  nearer  and  nearer  to  its 
edge ; we  thus  conclude  that  the  further  from  the  axis 
the  pencil  strikes  the  lens  the  greater  is  the  displace- 
ment of  the  image  from  its  proper  position. 

If  the  diaphragm  is  placed  behind  the  lens  the 
curvature  is  in  the  opposite  direction,  the  line  being 
convex  to  the  axis  of  the  lens. 


ABEERATION 


163 


In  practical  work  we  do  not  often  have  to  photograph 
two  sets  of  lines  at  right  angles,  but  it  will  be  interest- 
ing to  see  the  effect  produced  on  such  a set  by  the 
distortion;  Fig.  54  (1)  represents  the  object,  a square 
grating;  the  image  of  this  formed  with  the  diaphragm 
in  front  of  the  lens  is  given  in  Fig.  54  (2),  and  the 
image  when  the  diaphragm  is  behind  the  lens  in  Fig. 

54  (3);  in  the  last  two  diagrams  the  dotted  lines  show 
the  size  of  the  image  when  not  distorted. 

It  should  be  noticed  carefully  that  in  (2),  which  is 
called  barrel-shaped  distortion  the  image  is  smaller 
than  it  should  naturally  be,  and  also  that  the  curvature 
is  caused,  not  by  the  lines  being  bulged  outwards  at 
their  middle  points,  but  by  every  point  of  the  line 
being  displaced  inwards,  the  ends  of  the  lines  more  so 
than  the  middle  points. 

It  thus  appears  that  the  effect  of  the  distortion  is  to 
displace  every  point  towards  the  centre  of  the  picture, 
those  points  furthest  away  being  displaced  most ; the 
parts  of  the  picture  at  the  edge  are  therefore  unduly 
crowded.  On  the  other  hand,  in  (3),  sometimes  called 
pin-cushion  distortion,  the  effect  is  just  the  opposite ; 
every  point  of  the  picture  is  displaced  outward  from 
its  true  position,  those  points  at  the  edge  being  most 
displaced,  causing  a spreading  out  instead  of  a crowding 
of  the  edge  of  the  picture. 

This  view  of  the  phenomenon  is  illustrated  in  Fig. 

55  ; here  a series  of  circles  (1)  at  equal  distances  apart 
are  (2)  crowded  towards  the  centre  when  the  diaphragm 
is  before  the  lens,  being  most  crowded  at  the  edge, 
while  (3)  when  the  diaphragm  is  behind  the  lens  the 
circles  are  extended,  the  extension  being  most  marked 
at  the  edge ; in  (2)  and  (3)  the  dotted  circle  represents 
the  true  size  of  the  outer  circle. 

The  diagrams  are  very  much  exaggerated  to  bring 
out  clearly  the  points  they  illustrate. 

It  should  be  remarked  that  in  (2)  the  displacement 
of  different  circles  towards  the  centre  is  not  propor- 


164 


PHOTOGRAPHIC  OPTICS 


tional  to  their  radii,  for  then  the  effect  would  be  to 
reduce  the  size  of  the  diagram  merely  without  altering 


Fig.  55. 


the  relative  spacing  of  the  circle,  and  no  distortion 
would  be  produced ; a straight  line  would  then  remain 


ABERRATION 


165 


a straight  line,  but  would  be  shortened.  Similar 
remarks  will  apply  to  the  magnification  of  the  different 
circles  in  (3). 

The  phenomenon  of  distortion  is  seen  when  a large 
magnifying  glass  is  held  near  such  a rectangle  as  that 
in  Fig.  54  (1);  the  iris  of  the  eye  here  acts  as  the 
diaphragm,  and  the  appearance  is  that  of  Fig.  54  (3) ; 
if  instead  a diverging  lens  be  used  the  appearance 
produced  would  be  that  of  Fig.  54  (2). 

78.  Cause  of  the  Distortion. — It  is  usually  stated  in 
a vague  way  that  the  distortion  is  due  to  the  difference 
between  the  thickness  of  the  glass  traversed  by  rays  at 
the  centre  and  at  the  edge  of  the  lens ; but  by  the  aid 
of  the  properties  of  small  pencils  a more  detailed  and 
satisfactory  account  can  be  given,  though  a complete 
algebraical  proof  is  beyond  our  present  range.  We 
must  first  return  to  the  question  of  perspective  already 
treated  in  | 63  (Fig.  42). 

We  saw  that  according  to  the  elementary  theory  the 
picture  retains  its  proper  relative  proportions  if  an 
object  has  an  image  lying  on  the  secondary  axis  passing 
through  the  object. 

In  Fig.  42  take  for  example  the  object  A,  A 
is  the  line  joining  it  to  the  nodal  point  of  incidence, 
and  the  image  a lies  on  the  line  through  N2,  the  nodal 
point  of  emergence,  parallel  to  A N^. 

Similarly  parallel  respectively  to 

B Nj,  C N2,  etc. 

In  this  case  the  various  lengths  in  the  picture  have 
the  same  ratio  to  each  other  as  the  corresponding 
lengths  in  the  object  have  ; for  instance 
Fc:F(i=XB:XA. 

If,  however,  it  should  happen  that  the  images  of  A, 
B,  etc.  do  not  lie  at  a b,  etc.,  but  at  points  either 
nearer  to  or  further  from  the  axis  than  these,  and  at 
distances  from  it  not  proportional  to  F a and  F b,  then 
the  lengths  in  the  picture  will  no  longer  be  proportional 
to  those  in  the  object,  and  distortion  will  arise. 


166 


PHOTOGRAPHIC  OPTICS 


We  have  therefore  to  find  the  reason  why  the  lines 
joining  points  in  the  image  to  the  nodal  point  of 
emergence  are  not  parallel  to  the  lines  joining  the 
corresponding  points  of  the  object  to  the  nodal  point 
of  incidence ; and  also  to  show  that  the  deviation  is 
such  as  will  produce  the  effects  already  described.  As 
the  object  is  usually  distant  let  the  incident  ]3encil 
from  any  point  of  the  object  be  taken  to  consist  of 
parallel  rays  (this  will  simplify  future  work  without 
altering  the  nature  of  the  phenomenon). 

Let  A B (Fig.  56)  be  the  aperture  in  the  diaphragm, 
and  let  the  rays  in  the  plane  of  the  paper  meet  after 
refraction  at  P. 

Let  and  N2  be  the  nodal  points  of  incidence  and 
emergence,  and  X parallel  to  the  incident  ray;  draw 
N2  Y parallel  to  X 

Then  the  focal  line  perpendicular  to  the  plane  of  the 
paper  is  at  P,  that  in  the  plane  of  the  paper  at  some 
point  Q,  which  experiment  shows  to  be  in  most  cases 
further  from  the  lens  than  P. 

For  central  oblique  pencils  the  position  of  P is  usually 
as  shown  in  the  figure  between  N2  Y and  the  axis  of 
the  lens. 

The  circle  of  least  confusion  is  at  C about  midway 
between  P and  Q ; this  point  may  be  taken  as  the 
image.  Draw  lines  through  P and  Q perpendicular  to 
the  axis  to  meet  X2  Y in  R and  S,  and  C D parallel  to 
them  to  meet  Y in  D. 

Then  the  image,  instead  of  being  on  X2  Y as  it 
should  be  by  the  elementary  theory,  is  at  a distance 
C D below  it,  and  the  picture  is  therefore  crowded 
towards  the  centre,  as  it  should  be  when  the  diaphragm 
is  in  front  of  the  lens. 

If  we  know  the  positions  of  P and  Q we  can  find 
C D,  for  2 C D = P R + Q S. 

The  form  of  the  expression  for  the  aberration  shows 
that  it  is  proportional  to  the  square  of  the  distance 
from  tlie  axis  at  which  the  pencil  strikes  the  lens,  and 


ABEKRATION 


167 


in  Fig.  56  it  is  evident  that  the  distances  of  different 
points  from  the  axis  are  very  nearly  proportional  to 


168 


PHOTOGRAPHIC  OPTICS 


the  distances  from  the  axis  at  which  the  corresponding 
pencils  strike  the  lens.  Now  the  expression  for  the 
aberration  is  the  basis  of  all  calculations,  and  by  means 
of  it  together  with  that  for  tan  77,  where  77  is  the 
inclination  of  the  emergent  ray  from  the  axis,  we  can 
solve  any  problem  we  wish. 

The  first  power  of  does  not  occur  in  either  of  the 
expressions,  and  none  of  the  processes  of  the  calculation 
can  introduce  it. 

We  conclude  therefore  that  the  expression  for  any 
small  variation  from  the  elementary  theory  contains 
not  the  first,  but  the  second  power  of  ^ ; hence  C D is 
not  proportional  to  2;,  but  to 

We  see  therefore  that  the  distortion  C D diminishes 
the  distance  of  every  point  from  the  axis,  and  also 
that  it  is  proportional  to  the  square  of  the  distance  of 
the  object  from  the  axis  (higher  powers  of  being 
neglected). 

We  have  therefore  got  all  the  particulars  necessary 
for  the  explanation  of  the  barrel-shaped  distortion. 
We  shall  take  the  quantity  C D as  the  measure  of  the 
distortion,  reckoning  it  positive  when  C is  further  from 
the  axis  than  D (in  Fig.  56  C D is  negative). 

The  explanation  of  distortion  of  the  opposite  kind, 
when  the  diaphragm  is  behind  the  lens,  is  very  similar 
to  that  already  given,  and  can  be  understood  from  an 
inspection  of  Fig.  57,  which  is  lettered  to  correspond 
with  Fig.  56  ; here  P lies  on  the  side  of  N2  Y remote 
from  the  axis  of  the  lens. 

79.  Calculation  of  Distortion.  Numerical  Example. 

— As  the  subject  of  distortion  does  not  appear  to  have 
been  fully  treated  except  in  Coddington’s  Optics,  which 
is  long  since  out  of  print,  we  proceed  to  give  a 
numerical  example  which  will  to  some  extent  serve  as 
a justification  for  the  statements  made  in  the  last 
article. 

Take  the  case  of  a plane  convex  lens,  with  its  plane 
face  turned  towards  the  light,  for  which  the  calculations 


^foca/  Lme 


ABERRATION 


169 


Fig.  57. 


are  much  simplified  ; for  here  the  optical  centre  and 
therefore  the  nodal  point  of  emergence  are  at  the  point 


170 


PHOTOGRAPHIC  OPTICS 


where  the  curved  surface  meets  the  axis  (§  49,  end). 
Let  the  dimensions  of  the  lens  be  as  follows,  where  the 
symbols  have  the  meanings  previously  (§  49)  given  to 
them. 

1 r = CO  or  1/r  = 0,  s = 3 in.,  e = *2  in.  /a  = I “5. 

If  u be  the  distance  of  the  object  from  the  front 
surface,  and 'd  the  distance  of  the  image  from  the  back 
surface;  e the, inclination  of  the  incident  ray,  77  of  the 
emergent  ray  to  the  axis,  the  formulae  of  § 65  {Note) 
become,  since  Ijr  = 0 — 

1 _ I ya  - I e B , C \ 

2 o \ I 9 ) 

V u s fx,  ir  1 \ u u^] 

tan  rj  j ^ (jn  — 1)  fl  ~ 

tan  e vV.  u 2 /m  s \s  vJ _ ^ 


substituting  the  values  of  jn  and  s we  get 
1 1 ...  *133  H52 

u 


_ = i _ -167  - ^ - ^2  (-021  - 

V u \ 


+ 


•36i\ 

7^2 


tan  Tj 
tan  € 


u 

V L 


1 + 


•133 


2!' 

fs 


.2  / 1 

333  — 

'y, 


As  it  is  impossible  to  give  every  step  of  the  substi- 
tution, the  reader  should  verify  these  formulae  for 
himself.  They  are  the  fundamental  formulae  for  the 
lens,  and  by  means  of  them  we  can  find  the  position 
of  the  emergent  rays. 

Let  X be  the  middle  point  of  the  aperture  in  the 
diaphragm  (Fig.  58),  A and  N2  the  front  and  back 
surfaces  of  the  lens ; then  N2  is  the  nodal  point  of 
emergence. 

Take  the  pencil  parallel  to  X H such  that 

A X = I inch,  A H = *5  inch,  or 
u = \ inch,  2:  = ‘5  inch. 


^ From  the  mathematical  point  of  view  a flat  surface  may  be 
regarded  as  the  surface  of  a sphere  of  infinite  radius  or  of  zero 
curvature. 


ABEREATION 


171 


Let  the  second  ray  taken  be  K Y parallel  to  X H 
where  X Y = *2  inch ; then  by  similar  triangles 


172 


PHOTOGRAPHIC  OPTICS 


KH  :XY  = H A:  AX  = 1-2  KH  = 1 inch; 
2^'  = A Y = 1*2  inch,  2;'  = K A = *6  inch. 

Let  P U and  P Y be  the  two  emergent  rays  and 
angle  P U A = 77,  angle  P Y A = if,  also  angle  H X A 
- e;  then  ta7i  e = H A/A  X - 1/2  e = 26°  34'. 
For  the  ray  H X we  have 

1=  1 - -167  - -133  - 1(-021  - -152  + '361) 

V 4 


= 1 - *300  - *057  = *643 
V = 1*55  inch. 


= -643  [ 1 + -133  - A (-333  - -643) 
tan  e L ' 72  ^ \ 


= -643  [1  + -133  + -004]  = -7311  ; 
tany]  = ‘3655,  rj  = 20°  5'. 

For  the  ray  K X we  have  u'  = 1 *2  in.,  z = *6  in. 


= *833  - *167  - *092  - *052  = *522 
.*.  v'  = 1*91  inch. 


= *626  [1  + *111  + *004]  = *6980; 
tan  77'  ==  *3490  77'  = 19°  14'. 

Xow,  PM  = PYsmPYM  = 


= 2*745  inches. 

Also  it  can  easily  be  shown  that 


YM  = (v  - v)- 
= 7*868“  inches. 


sin  (77  — 77')  sin  51' 


sin  Tj  cos  Tj'  sin  20°  5',  cos  19°  14 

_ — ^ ^ ^ 


ABERRATION 


173 


KgM  = YM  - y]sr2  = 7*87  - 1*91  = 5*96  in. 
Let  the  perpendicular  from  P on  the  axis  meet  the 
secondary  axis  through  1^2  ^ i then — 

R,M  = N2  M R ^2  M = N2  M ta7i  H X A = 5 96 
X *5  = 2*98  inches. 

.*.  PR-RM-PM  - 2*98  - 2*74  - *24  inch. 


We  have  thus  found  the  distance  below  the  secondary 
axis  of  the  focal  line  perpendicular  to  the  plane  of  the 
paper. 

We  must  now  find  the  position  of  the  other  focal  line. 

The  incident  rays  being  refracted  at  a plane  surface 
will  be  parallel  after  refraction,  and  we  have  a pencil 
of  parallel  rays  striking  the  curved  surface ; hence 
(§  73a)  the  focal  line  required  will  be  at  the  point 
where  a line  through  O,  the  centre  of  the  surface, 
parallel  to  the  pencil  striking  the  curved  surface,  meets 
the  mean  emergent  ray  U P.  Let  Q be  this  point. 

Let  0 and  (j)  be  the  angles  of  incidence  and  refraction 
at  the  plane  surface,  then — 


tan  6 = I,  .*.  0 = 26°  34' 
sin  0 sin  26°34' 

""  PS 


.*.  (/>  = 17°  21' 


Hence  the  angle  Q 0 M must  be  17°  21'. 

Draw  Q L perpendicular  to  the  axis,  and  produce  it 
to  meet  X2  R in  S ; then — 


QL  = OU 


sin  P U A . sin  Q O M 
sin  U Q O 

, ^^sin  20°  5'  . sin  17°  21' 

1 *45  

sin  2°  44' 


3*08  inches. 


LU  = OU 


cos  P U A . sin  Q O M 


1*45 


sin  U Q O 

C05  20°  5'  17°  21' 

sin  2°  44' 


8*53  inches. 


.*.  LX2  = LU  - N2U  = 8*53  - 1*55  = 6*98  inches. 


174 


PHOTOGRAPHIC  OPTICS 


Then,  S L = L ^2  S N2  L = L ISr2  H X A = 
6*98  X *5  = 3*49  inches. 

.*.  QS  = SL~QL  = 3*49  - 3*08  = *41 

If  C be  the  circle  of  least  confusion  midway  between 
P and  Q,  and  C T be  drawn  perpendicular  to  the  axis  to 
meet  X2  R on  T,  we  get 

C T = -t  (P  R + Q S)  = (-24  + *41)  = *32  inch. 

.*.  Distortion  = — *32  inch. 

The  principal  focal  length  of  the  lens  is  6 inches, 
hence  the  principal  focus  P is  very  near  to  M on  the 
side  remote  from  the  lens. 

Mathematical  readers  will  easily  find  that  the  image 
is  curved  away  from  the  lens,  and  that  its  radius  of 
curvature  is  very  nearly  9 inches. 


II. — Chromatic  Aberration 

80.  We  have  seen  (§  16)  that  when  white  light  is 
refracted  raj^s  of  light  of  different  colours  are  differently 
deviated,  the  deviation  being  greatest  for  violet  and 
least  for  red  rays  ; and  the  angle  between  a standard 
ray  and  any  other,  after  refraction,  was  called  the 
dispersion  of  that  ray. 

Since  lenses  act  by  refraction,  they  will  exhibit  the 
phenomenon  of  dispersion,  with  the  result  that  the  rays 
of  different  colours  will  not  always  come  to  a focus  at 
the  same  point. 

This  can  be  seen  from  the  consideration  of  the 
expression  for  the  focal  length  of  a lens — 


In  this  /X,  the  refractive  index,  is  different  for  different 
rays,  being  greater  for  violet  than  for  red,  showing  that 
the  focal  length  is  less  for  violet  than  for  red  rays. 

The  effect  of  this  is  that  a single  lens  does  not  form 
one  picture,  but  several  of  different  colours  at  different 


ABERRATION 


175 


distances,  the  violet  one  being  the  nearest  to  the  lens. 
This  can  be  easily  tested  by  exploring  the  cone  of  rays 


formed  by  a totally  uncorrected  lens,  when  white  light 
from  a point  on  its  axis  falls  on  it. 


176 


PHOTOGRAPHIC  OPTICS 


If  the  screen  be  placed  near  the  lens  the  circular 
patch  of  light  is  white  in  the  middle,  but  edged  with 
red  ; as  the  screen  is  moved  further  away  the^ patch 
contracts,  and  careful  observation  shows  evidences  of 
the  foci  for  the  different  colours  ; and  beyond  the  foci 
the  circular  patch  is  white  in  tlie  centre,  but  edged 
with  violet. 

An  examination  of  Fig.  59  will  show  that  this  is 
what  we  should  expect.  The  cones  formed  by  the 
violet  and  red  rays  are  shown ; Y is  the  geometrical 
focus  for  violet  light,  II  that  for  red  light. 

For  points  nearer  to  the  lens  than  Y,  the  cone  of 
red  rays  is  the  larger,  giving  a white  patch  on  the 
middle  where  the  colours  overlap  with  an  edge  of  red  ; 
beyond  R on  the  other  hand  the  violet  cone  overlaps, 
giving  a white  centre  with  violet  margin. 

The  state  of  affairs  between  Y and  R cannot  be 
accurately  stated,  as  it  is  complicated  by  spherical 
aberration,  but  careful  observation  with  a particular 
large  lens  showed  that  when  the  screen  was  moved 
away  from  the  lens  through  Y and  R,  small  discs  of 
various  colours  appeared  on  it. 

In  this  description,  for  the  sake  of  simplicity  two 
colours  only  have  been  considered,  but  the  reader  will 
have  no  difficulty  in  imagining  the  actual  state  of 
affairs  when  the  intermediate  rays  are  taken  into 
account ; it  will  not  be  very  different  from  that 
described. 

In  the  early  days  of  photography  most  of  the  lenses 
used  showed  Chromatic  Aberration,  and  as  the  rays 
affecting  the  eye  are  different  from  those  most  effective 
on  the  photographic  plate,  it  was  necessary,  after 
obtaining  the  visual  focus,  to  give  the  plate  a slight 
shift  to  make  the  resulting  picture  sharp. 

This  was  of  course  an  inconvenience,  but  so  long  as 
plates  were  sensitive  only  to  a very  small  part  of  the 
solar  spectrum  it  was  not  an  insuperable  objection  ; but 
now  that  plates  have  been  made  sensitive  to  a much 


ABERRATION 


177 


wider  range  of  rays  the  adjustment  will  no  longer  do 
what  is  required. 

81.  Irrationality  of  Dispersion. — The  problem  in 
hand  is  to  destroy,  by  the  use  of  two  or  more  lenses, 
the  dispersion  of  the  various  rays,  without  at  the  same 
time  destroying  their  deviation  ; or  in  other  words  to 
make  the  rays  of  all  colours,  coming  from  one  point  of 
the  object,  coincide  after  their  passage  through  the  lens 
without  at  the  same  time  destroying  the  converging  or 
diverging  power  of  the  lens.  Newton,  misled  by  an 
experiment,  believed  that  this  was  impossible,  and  his 
opinion  being  generally  accepted,  long  hindered  the 
improvement  of  lenses,  and  turned  the  attention  of 
opticians  from  refracting  to  reflecting  telescopes. 

The  mistake  was  discovered  by  Dollond,  who  produced 
the  first  achromatic  lenses ; it  is  said  to  have  been 
previously  discovered  by  a Mr.  Hall  of  Worcester,  but 
this  is  doubtful. 

To  understand  the  difficulty,  consider  a ray  of  light 
passing  successively  through  two  prisms  of  the  same 
material  and  of  equal  angles,  turned  in  opposite 
directions  with  their  adjacent  faces  parallel  (Fig.  60);^ 
let  the  ray  of  white  light  P Q fall  on  the  first  prism, 
and  let  S S'  and  T T'  be  the  emergent  red  and  violet  rays 
respectively;  the  angles  of  incidence  of  these  rays  on 
the  second  prism  will  be  the  same  as  their  angles  of 
emergence  from  the  first.  The  second  prism  will 
therefore  produce  in  each  ray  a deviation  equal  to  that 
caused  by  the  first  prism,  but  in  an  opposite  direction, 
and  the  rays  will  emerge  parallel,  but  parallel  also  to 
P Q,  and  though  the  dispersion  is  corrected,  the  deviation 
is  destroyed  also. 

The  ratio  of  the  deviation  to  the  dispersion  is  the 
same  in  both  lenses,  so  that  whenever  we  destroy  the 
one  we  also  destroy  the  other. 

The  mistake  Newton  made  was  thinking  that  the 
ratio  of  dispersion  to  deviation  is  the  same  for  all 

^See  Glazebrook’s  Physical  Optics,  2nd  Ed.,  p.  219. 

N 


178 


PHOTOGRAPHIC  OPTICS 


substances.  Dollond  showed  that  by  using  two  prisms 
of  different  materials,  for  instance  of  crown  glass  and 


ABERRATION 


179 


flint  glass,  the  dispersion  could  be  destroyed  but  a 
considerable  deviation  left. 


180 


PHOTOGRAPHIC  OPTICS 


Flint  glass  has  a much  higher  ratio  of  dispersion  to 
deviation  than  crown  glass  ; a prism  of  crown  glass  of 
an  angle  60°  will  produce  the  same  dispersion  between 
the  red  and  violet  rays  as  a prism  of  flint  glass  with  an 
angle  of  37°,  but  the  deviations  are  by  no  means  the 
same  in  the  two  cases. 

If  prisms  of  crown  and  flint  glass  be  arranged  to 
produce  spectra  placed  one  over  the  other,  for  comparison 
(Fig.  61),  so  that  the  lines  C and  H of  the  solar  spectrum 
coincide,  the  remaining  portions  of  the  spectra  will  not 
be  found  to  be  at  all  identical ; for  instance,  the  lines 
D and  E in  each  will  not  coincide.  Thus,  if  two  lines 
of  the  spectra  coincide  the  intermediate  lines  do  not  do 
so  too. 

This  phenomenon  is  called  the  Irrationality  of 
Dispersion. 

The  effect  of  this  irrationality  is,  that  although  the 
rays  of  two  particular  colours  may  be  made  to  have 
the  same  deviation,  yet  rays  of  other  colours  will  have 
different  deviations,  and  the  emergent  light  will  be 
tinted  and  exhibit  what  is  called  a Secondary  Spectrum. 

If  three  prisms  be  used,  rays  of  three  different 
colours  may  be  made  to  coincide,  and  the  emergent 
light  will  be  much  less  coloured  than  when  two  only 
are  employed. 

Coddington  in  his  treatise  on  optics  gives  a table 
(p.  181)  of  refractive  indices  for  different  lines  of  the 
Solar  Spectrum  extracted  from  a paper  of  Fraunhofer, 
and  this  will  serve  as  an  illustration  of  the  statements 
made  above. 

The  absence  of  regularity  in  the  dispersion  of  these 
substances  is  illustrated  by  Coddington  in  the  following 
table  (p.  182),  which  contains  the  differences  of  the 
numbers  in  the  last,  exhibiting  the  intervals  between 
the  fixed  lines  in  the  several  spectra. 

The  last  column,  it  should  be  noticed,  gives  the 
difference  between  the  refractive  indices  for  the  extreme 
lines,  B and  H,  considered. 


ABERRATION 


181 


182 


PHOTOGRAPHIC  OPTICS 


Refracting  Medium 

^ — 1 

BC 

CD 

DE 

EF 

FGDH 

BH 

Water 

3309 

8 

19 

22 

20 

35 

29 

133 

Solution  of  Potash . 

3996 

9 

23 

28 

25 

45 

38 

168 

Spirit  of  Turpentine 

4705 

10 

29 

39 

34 

65 

47 

234 

Crown  Glass  . .13 

5243 

10 

27 

34 

29 

46 

48 

204 

„ . . 9 

5258 

10 

28 

34 

30 

56 

49 

207 

„ . . M 

5548 

11 

32 

41 

36 

68 

59 

i 246 

Flint  Glass  . . 3 

6020 

18 

47 

60 

55 

108 

96 

384 

„ . . 30 

6236 

19 

51 

67 

62 

119 

106 

424 

„ . . 23 

6266 

18 

52 

69 

63 

120 

109 

1 431 

„ . . 13 

6277 

20 

53 

70 

62 

121 

107 

433  i 

The  action  of  prisms  has  been  considered  here  in 
place  of  that  of  lenses,  for  the  nature  of  the  phenomenon 
is  similar  in  the  two  cases,  and  prisms  are  easier  than 
lenses  to  think  about. 

82.  Chromatic  Aberration  of  a Thin  Lens. — In  this 
case  we  shall  consider  only  a central  pencil  of  rays, 
parallel  to  the  axis  ; we  have  seen  that  the  principal 
focal  lengthy  of  the  lens  is  given  by 


Now  let  fjii  and  /X2  be  the  refractive  indices  for  red  and 
violet  rays  respectively,  and  let  and  be  the  corre- 
sponding focal  lengths,  then 


- (Mi-1) 

For  brevity  denote 


1 ^ 


- - )and-  = (^2  - 1)(_  - - 


1 


— - by  - 
r 8 p 


•••/i  =i 


P f - P 
“ 


/Xi  - i - M2  - 1 

.*.  The  distance  between  the  two  foci,  or  the  chromatic 
aberration  of  the  lens,  is 


A -A  = p 


1 

Ml-  i 


M2 


1 


= p 


M2 Ml 


{P'1—  1)  (m2  1) 


ABERRATION 


183 


Now  if  fjL  represent  the  mean  value  of  the  index  of 
refraction,  we  may  without  serious  error  assume  that 

(a*!””  1)  (a*2““  1)  = (a*  — 1)^ 

Hence 


Jl  ^^2 


_ ^ ^ _ 


The  quantity 


(f.1  —If  fX—  1 
H'2  — 


/^2  — ^1 

/i  - 1 


- 1 


is  called  the  Dispersive  Power 


- 1 


of  the  medium,  and  is  often  denoted  by  w,  and 

evidently  ^ where  f is  the  mean  focal  length. 

Hence  we  get 

Chromatic  Aberration  = Mean  focal  length  X 
dispersive  power, 

or/i  - /a  = «/• 

This  expression  is  of  great  importance. 

The  quantity  w,  the  dispersive  power  of  the  substance, 
is  the  form  in  which  the  difference  between  the  re- 
fractive indices  for  the  different  rays  enters  into  the 
calculations. 

83.  Chromatic  Aberration  for  a Thick  Lens. — In  this 
case  the  calculation  is  not  quite  so  simple  ; and  we 
must  here  remember  that  the  positions  of  the  nodal 
points  depend  on  the  refractive  index,  and  hence  vary 
for  different  rays. 

We  must  therefore  measure  our  distances  from  the 
surfaces  of  the  lens  ; let  E be  the  distance  of  the  princi- 
pal focus  from  the  back  surface,  and  let  the  symbols 
have  their  usual  meanings  (§  44)  ; then 


~ ~ 1)  (r“  b “ 


(m  - 1)^ 


\r  Sj 

From  this  it  can  be  proved,  that  if  w is  the  dis- 
persive power,  the  value  of  the  chromatic  aberration  is 
r . e2  (^  - 1)3) 


184 


PHOTOGRAPHIC  OPTICS 


Examples. — Find  the  Chromatic  Aberration  for  a 
thin  lens  where  r = — 7 inches,  = 5 inches,  /x  = 
1*524,  w = *0102  ; also  for  a lens  of  thickness  e = '2 
inch. 

(a)  Thin  lens. 

Here^  = („  - 1)(;  - })  = '524  (-i  - i) 

= -*179,  .*./  = - *5*58  inches. 

.*.  Aberration  — (h  f = — *0102  X 5*58  = — ‘057 
inch. 


(6)  Thick  lens. 


Here  | 


}]-  ^ ~ 

s J ^ 


*179 


- *001  = *180  .*.  E = - 5*56 


.*.  Aberration  = *01021  — 5*56  + 


*0102 


{- 


{- 


5-56  + -007 


•2  X 30-9  X -US'! 
2-39  X 49  / 

= — *056  inch. 


Comparing  these  results  we  see  that  they  differ  only 
by  one  thousandth  of  an  inch,  and  thus  the  aberration 
is  practically  the  same  for  both  cases. 

When  e the  thickness  is  small  compared  with  the 
radii  of  curvature  of  the  faces  we  may  neglect  its  effect 
and  take  the  aberration  to  be  the  same  as  that  for  a 
thin  lens  with  the  same  radii  of  curvature,  with  a 
sufficient  approach  to  accuracy  for  all  practical  purposes. 

84.  Condition  that  two  Lenses  in  Contact  should 
form  an  Achromatic  Combination. — Let  be  the 

radii  of  the  first  lens,  and  p-^'  its  refractive  indices 
for  the  two  rays  which  are  to  be  combined,  and  let 
^?2,  1^2  be  the  corresponding  quantities  for  the  second 

lens,  and  let 


1 

Pi 


1 


1 1 

^*1  ’ P2 


‘1 


ABERRATION 


185 


Then  the  focal  lengths  of  the  lenses  are  (§82) 


- 1 


and 


P-i 


^ /^2  - 1 
and  (§  40)  the  focal  length  F of  the  combination  is 
given  by 

^ ^ _i_  f^2  — ^ 

F Pi  P2 

Let  u and  v be  the  distances  of  object  and  image 
from  the  lens  for  rays  of  both  colours;  we  therefore  get 

1 

V 


1 = Ih-A  + also 


Pi 


P2 


1 - i = + jVjlJ 

V U p^ 

Hence  we  must  have  for  achromatism 


P-i  1 Pi  ^ P2  ^ (^2 ^ _ 

Pi  Pi  P2  P2 

or  = 0 

Pi  P2 

Let  ^2  be  the  mean  refractive  indices,  then 
.»/  - f^l  X ^'1  ~ ^ + t^-2  ~ ^‘-2  . _j_  ~ 1 = 

;’i  - 1 ■ Pi  1\,  - 1 p.2 

and  are  the  mean  focal  lengths  this  becomes 


^ +^?  = 0 

Ji  J2 


(a) 


CO2  being  as  before  the  dispersive  powers  of  the  glass 
of  which  the  lenses  are  made. 

This  determines  the  ratio  tlie  mean  focal 

lengths  of  the  lenses ; if  the  focal  length  of  the  required 
lens  is  to  be  F,  we  have  also 


F 


= 1 +-' 
A J\ 


(0 


186 


PHOTOGRAPHIC  OPTICS 


The  values  of  and  can  then  be  found  from  {a) 
and  (h)  by  elementary  algebra. 

It  is  worthy  of  notice,  that  the  Chromatic  Aberration 
is  by  this  arrangement  corrected  for  rays  from  points  at 
all  distances  from  the  lens. 

Example. — Let  it  be  required  to  find  the  focal  lengths 
of  two  lenses  composed  of  Crown  Glass  No.  13  and 
Flint  Glass  No.  13  (see  table,  § 81),  taking  the  extreme 
rays  as  B and  G,  and  E as  the  mean  ; the  lens  to  be 
converging  and  of  6 inches  focal  length. 


Here,  = 1*5243,  = 1*5314,  p/  = 1*5399. 

/X2  = 1-6277,  ^2  = 1*6420,  = 1*6603. 

- -0102 


= -0198 


t'l 

“ 1-5314 

l - 1^  -2  , 

•0326 

i'2 

“ 1-6420 

•0102 

•0198 

■— r—  + 

= 

/i 


/2 


A 

or  y = — 
■'2 


•0102 

•0198 


= - -515 


• («) 


Also,  since  the  focal  length  of  the  combination  is  to 
be  6 inches,  we  must  have 


w 


which  give  on  solution 

— 2*91  inches, /2  = 5*64  inches. 

This  shows  that  the  Crown  Glass  lens  must  be  con- 
verging, and  the  Flint  Glass  lens  diverging. 

85.  Achromatism  of  three  or  more  Lenses  in  Contact. 
— Let  the  mean  focal  length  of  the  lenses  bey*^ 
their  dispersive  powers  co^,  CO2,  ^3?  then  if  F be  the  focal 
length  of  the  combination 


ABEREATION 


187 


and  the  condition  for  achromatism  can  be  shown  much 
in  the  same  way  as  before  to  be 

Wo 


^ 4.  " 


/l  ^2 


+ -^Hh  = 0 . 


(«) 


We  have  two  equations  as  before,  but  tliey  are  not 
enough  to  determine  completely  the  values  of  f.2^  /g, 
etc. ; hence  we  can  introduce  other  conditions  till  we 
get  enough  relations  to  determine  the  focal  lengths 
completely. 

For  instance,  if  we  have  three  lenses,  we  may  use 
them  to  combine  three  rays,  suppose  B,  E,  and  G.  Let 
Wj,  (S3  be  the  dispersive  powers  for  rays  B and  E, 
and  <Sj^\  (S2^,  for  rays  E and  G. 

The  condition  for  the  combination  of  rays  B and  E 
is 


^ + t^i+‘-5^  = o 

/l  /2  Jz 

and  that  for  the  rays  E and  G is 


wd  w,/  w 
— 4 -I 

fi  A A 


n j% 

and  if  the  lens  is  to  have  focal  length  F, 


(a) 

{h) 


F 


1 


1 


fiA  A 


These  three  equations  on  solution  will  give  the  values 

86.  Achromatism  of  Lenses  not  in  Contact. — Let 

be  the  mean  focal  lengths  of  the  lenses,  e their  dis- 


tance apart ; then  it  can  be  shown 
for  achromatism  is^ 


that  the  condition 


+ — - 

A A 


or  ^ — 


(A  + 


(A  + ' /1/2  “2  AAz 

If  two  lenses  of  the  same  kind  of  glass  be  employed, 
then  w^,  and  the  relation  becomes 


^ Coddington,  p.  254. 


188 


PHOTOGEAPHIC  OPTICS 


(4  + .)2^  -4/2 

From  this  e = — fi  ^ 

We  can  thus  achromatize  two  lenses  of  the  same  kind 
by  placing  them  at  a suitable  distance  apart,  but  this 
distance  will  not  be  usually  convenient  in  practice. 

There  is  also  another  disadvantage  in  this  case,  that 
the  images,  though  being  at  the  same  place,  may  not  be 
of  the  same  size. 

87. — It  will  be  interesting  to  the  mathematical  reader 
to  note  that  most  of  the  relations  given  can  be  at  once 
obtained  by  differentiation ; for  instance,  take  the  rela- 
tion of  § 50. 

Here,  using  the  notation  of  that  article,  we  have 


1 ^ , M2  — 1 

F “ Pi  P2 

The  condition  is  that  F shall  not  vary  for  variations 
of  F^  and  Fg  or  F = 0 j hence  differentiating  with 
respect  to  and  we  get 

d Ml  ^ Q 

Pi  P2 


or 


Since 


V 


Pi 


d 

- 1 


- 1 


Pi 
, and 


+ 


d lJi.2  V.2 


- 1 


P2 


= 0 


P2_ 

^r-^T 


, this  skives  as  before 


^ I ^ 

1 J'2 


■=  0 


88.  Combined  Effect  of  Chromatic  and  Spherical 
Aberrations. — Consider  an  oblique  small  pencil  through 
an  uncorrected  lens ; on  refraction  there  will  be  a small 
pencil  of  each  colour,  each  with  its  focal  lines.  Think- 
ing, for  instance,  of  two  colours,  red  and  violet,  it  some- 
times happens  that  one  red  focal  line  coincides  with  the 
violet  focal  line  at  right  angles.  This  gives  rise  to  a 
cross  with  two  points  red  and  two  points  white. 

This  may  be  practically  realized  by  placing  some 


ABERRATION 


189 


globules  of  mercury  on  a horizontal  plate,  illuminating 
them  by  a candle  or  lamp  in  a suitable  position,  and 
receiving  the  image  of  them  produced  by  a single  lens 
on  a ground  glass  screen. 

If  the  screen  be  moved  about,  the  position  which 
gives  the  two-coloured  cross  can  easily  be  found. 

The  experiment  is  well  worth  performing,  as  the 
appearance  is  one  of  great  beauty. 


CHAPTER  lY 


THE  CORRECTION  OF  ABERRATION  AND  THE  DESIGN  OF 
LENSES 

89. — The  correction  of  aberration  and  the  design  of 
a lens  are  subjects  which  it  would  be  hard  to  separate. 
In  the  chapter  on  aberrations  we  have  examined  various 
faults  in  lenses,  and  we  shall  now  reap  our  reward  in 
learning  how  these  faults  are  to  a great  extent  corrected. 

Let  us  think  what  would  be  the  main  points  for 
which  we  should  look  in  an  ideal  lens ; they  would  be 
somewhat  as  follows — 

(i)  There  would  be  no  spherical  aberration  at  points 
near  the  axis,  so  that  as  far  as  the  centre  of  the  picture 
is  concerned  a large  aperture  would  give  good  definition. 

(ii)  The  lens  would  be  achromatized  for  several 
colours. 

(iii)  The  definition  would  be  sharp  at  some  consider- 
able distance  from  the  centre  of  the  picture,  so  that 
both  the  astigmatism  for  oblique  pencils  would  be  small 
and  the  field  of  view  flat. 

(iv)  There  would  be  no  flare  spot. 

(v)  The  lens  would  be  rapid. 

(vi)  There  would  be  no  distortion. 

Such  a lens  as  this  cannot  be  realized,  for  some  of 
the  conditions  are  incompatible ; as  is  well  known  from 
experience,  (iii)  requires  a small  aperture,  and  this  is 
incompatible  with  (v). 

Again,  if  condition  (i)  is  satisfied  for  objects  at  one 
distance  it  will  not  in  general  be  satisfied  for  objects  at 

190 


THE  COREECTION  OF  ABERRATION' 


191 


another  distance,  which  reminds  us  that  in  designing  a 
lens  account  must  be  taken  of  the  purpose  for  which  it 
is  intended.  If  it  is  required  for  landscape  work,  then 
the  spherical  aberration  must  be  zero  for  objects  at  a 
very  great  distance,  but  if  it  is  to  be  used  for  por- 
traiture or  copying,  then  the  aberration  must  be  zero 
for  objects  at  a definite  small  distance. 

The  problem  of  the  design  of  lenses  was  first  attacked 
by  astronomers,  Huyghens  and  Newton  being  amongst 
the  earliest  in  the  field,  but,  as  mentioned  before  (§  81), 
Newton,  misled  by  an  experiment,  thought  it  was 
impossible  to  correct  chromatic  aberration,  and  in 
consequence  turned  his  attention  to  reflecting  telescopes. 

In  1747  Euler,  from  a study  of  the  eye,  concluded 
that  achromatism  was  possible,  and  suggested  that  in 
the  eye  it  was  due  to  the  various  densities  of  the  media 
which  compose  the  lens ; against  this  Dollond  quoted 
Newton’s  conclusion,  and  Euler  yielded  to  so  great  an 
authority.  However,  Klingenstierna,  a professor  of 
the  University  of  Upsala,  ventured  to  doubt  Newton’s 
results,  and  submitted  them  to  a careful  examination, 
proving  mathematically  that  they  would  lead  to  absurd 
consequences. 

Dollond  thereupon  repeated  Newton’s  experiments, 
and  found  that  although  the  conclusions  are  true  for 
one  particular  case,  yet  they  are  not  true  for  all  cases, 
or  in  other  words,  that  Newton  had  generalized  too 
hastily ; he  then  set  to  work  to  make  an  achromatic 
lens,  and  was  successful.  Since  that  time  a great  deal 
of  study  has  been  bestowed  on  the  subject,  and  the 
many  excellent  lenses  existing  at  the  present  day  are 
the  result. 

The  theory  of  the  design  of  astronomical  lenses  is 
not  altogether  sufficient  for  that  of  photographic  lenses ; 
in  a telescope  the  lenses  are  used  with  a very  small 
angle,  2°  or  3°,  but  in  photography  lenses  with  angles 
of  45°  or  60°  are  often  used,  and  some  embrace  even 
100°. 


192 


PHOTOGRAPHIC  OPTICS 


In  an  astronomical  lens  it  is  therefore  enough  to 
destroy  the  aberration  near  the  axis,  but  in  a photo- 
graphic lens  it  must  also  be  destroyed  at  some  distance 
from  the  axis,  and  also  the  field  of  view  must  be  fairly 
flat. 

90.  Definition  of  a Sharp  Image. — Before  studying 
the  correction  of  aberration  we  must  first  form  a clear 
idea  of  what  we  mean  by  a sharp  image ; this  has 
already  been  treated  in  connection  with  pinhole  photo- 
graphy, but  it  will  be  well  to  recall  it.  We  know  from 
experience  that  when  any  point  of  an  image  is  clearly 
focussed  on  the  ground  glass,  a small  displacement  of 
the  screen  can  be  made  without  producing  any  appre- 
ciable blurring  in  the  image.  We  have  therefore  a 
certain  latitude  in  focussing,  which  much  simplifies  the 
problem  in  hand. 

We  have  thought,  in  the  elementary  theory,  of  the 
image  of  a point  as  consisting  of  an  absolute  point,  but 
this  is  not  necessary  for  a sharp  picture ; if  the  image 
consists,  not  of  a point,  but  of  a patch  of  light,  whose 
largest  dimension  does  not  subtend  at  the  eye  an  angle 
greater  than  about  one  minute  when  the  picture  is 
held  at  a convenient  distance,  then  the  patch  is  not 
distinguishable  from  a point. 

The  size  of  the  patch  will  of  course  depend  on  the 
distance  at  which  the  picture  is  to  be  viewed,  and  must 
be  smaller  the  nearer  the  picture  is.  An  enlargement 
which  is  to  be  viewed  at  a distance  of  four  or  five  feet 
is  very  often  coarse  when  examined  closely. 

The  most  useful  distance  to  consider  is  that  of  the 
least  distance  of  distinct  vision  for  normal  sight,  for  if 
the  picture  then  appear  sharp  it  will  be  sharp  at  all 
other  distances  also ; we  have  seen  that  in  this  case  the 
greatest  permissible  dimension  is  about  ‘01  cm.  or 
‘004  in. 

Let  us  call  this  dimension  for  convenience  2 e,  so 
that  when  the  patch  is  circular  its  radius  is  e.  The 
effect  of  the  above  is  that  we  need  not  correct  the 


THE  COKRECTJON  OF  ABERRATION 


193 


aberration  completely  all  over  the  picture,  it  will  be 
enough  if  we  can  correct  it  for  points  near  the  axis, 
and  keep  it  within  certain  limits  for  points  distant  from 
the  axis ; or  in  other  words,  the  picture  will  appear 
sharp  if  the  aberration  is  corrected  for  points  near  the 
axis,  and  at  points  distant  from  the  axis  the  sections  of 
the  pencils  by  the  ground  glass  or  plate  have  their 
greatest  breadth  not  more  than  2 e. 

The  problem  in  hand  is  to  design  a lens,  or  com- 
bination of  lenses,  which  shall  be  corrected  for  aberra- 
tion ; this  can  be  done  by  either  of  two  methods,  one 
direct,  the  other  indirect.  We  shall  consider  for 
example  the  case  of  two  lenses. 

91.  Simple  Combination  of  two  Lenses.  — These 
combinations  consist  of  two  lenses  of  difierent  kinds  of 
glass,  one  concave  and  one  convex,  either  placed  at  a 
fixed  distance  apart  or  cemented ; the  convergent  lens 
is  usually  of  crown  glass,  the  divergent  lens  of  Hint 
glass.  The  conditions  which  the  lens  must  fulfil  are 
the  following — 

(a)  The  lens  must  have  a given  focal  length. 

(b)  The  spherical  aberration  for  rays  parallel  to  the 
axis,  or  for  rays  coming  from  some  definite  point  on 
the  axis,  must  be  corrected. 

(c)  The  chromatic  aberration  must  be  destroyed. 

In  addition  to  this  the  aberration  at  points  off  the 
axis  should  be  as  far  corrected  as  possible,  so  that  the 
lens  may  be  worked  with  an  aperture  as  large  as 
possible. 

We  shall  assume  that  the  glasses  for  the  two  lenses 
have  been  chosen,  and  that  their  refractive  indices  and 
dispersive  powers  have  been  found. 

There  are  then  four  quantities  to  be  determined  by 
calculation,  i,  e.  the  radii  of  the  four  surfaces ; the 
lenses  are  usually  in  contact  and  their  thickness  small 
enough  to  be  negligible. 

The  three  conditions  above  when  expressed  in  terms 
of  the  radii  give  three  equations,  which  are  not  enough 

o 


194 


PHOTOGRAPHIC  OPTICS 


by  themselves  to  determine  the  four  radii,  and  another 
condition  can  therefore  be  introduced  to  provide  the 
remaining  equation  required. 

The  condition  most  usually  imposed  is  that  of 
Clairaut,  who  made  the  radii  of  the  two  adjacent 
lenses  the  same,  so  that  they  could  be  cemented ; 
d’Alembert  proposed  to  use  it  to  improve  the  definition 
at  points  not  on  the  axis ; while  J.  W.  Herschel  used 
it  to  correct  the  aberration  for  rays  coming  from  a 
point  on  the  axis  at  a finite  distance,  as  well  as  for  rays 
parallel  to  the  axis. 

When  the  four  conditions  are  obtained  the  four  radii 
can  theoretically  be  found,  though  the  algebraical  work 
is  heavy.  ^ 

92.  Indirect  Method. — Another  method  is  some- 
times adopted ; a rough  approximation  to  the  lens 
required  is  taken  as  the  starting-point.  The  course  of 
a ray  parallel  to  the  axis,  and  cutting  the  lens  at  a 
given  distance  from  the  axis,  is  calculated  by  trigono- 
metry, the  ray  taken  being  intermediate  in  colour 
between  the  extreme  rays,  and  the  point  where  this  ray 
cuts  the  axis  is  found.  The  corresponding  point  for 
rays  very  near  the  axis  is  then  found ; if  these  two  cut 
the  axis  in  very  near  points,  similar  calculations  are 
then  made  for  rays  of  the  extreme  colours  used.  If 
all  the  points  thus  found  are  very  close  together  the 
lens  is  satisfactory ; if  not  the  computer  judges  what 
kind  of  changes  must  be  made  in  the  curvatures  of  the 
surfaces,  and  repeats  the  calculations. 

This  process  is  continued  until  by  successive  approxim- 
ations a satisfactory  result  is  arrived  at. 

This  method  appears  long  and  cumbersome,  but  it  is 
probably  not  so  arduous  as  it  appears,  for  a computer 
who  has  an  instinct  for  his  work  acquires  to  a remark- 
able degree  a knowledge  of  all  parts  of  the  calculation, 
which  enables  him  to  estimate  the  nature  of  the  change 


^ See  Martin’s  paper  quoted  above. 


THE  CORRECTION  OF  ABERRATION 


195 


in  the  result  which  various  changes  in  the  data  would 
produce. 

93.  Three  or  more  Lenses. — When  more  than  two 
lenses  are  employed  there  will  be  more  quantities  to  be 
determined,  thus  making  it  possible  to  satisfy  more 
conditions,  such  as  rendering  the  field  of  view  flat, 
correcting  aberration  at  points  not  on  the  axis,  and 
so  on. 

An  arrangement  recommended  by  M.  Martin  is  to 
make  the  nodal  points  of  the  lens  which  is  equivalent 
to  the  system  coincide,  so  that  the  combination  shall 
act  like  a single  thin  lens. 

The  details  of  the  calculations  are  very  intricate,  and 
can  be  properly  grasped  only  by  one  well  used  to  such 
work ; no  general  method  of  procedure  can  be  given, 
as  each  case  must  be  considered  on  its  own  merits. 

94.  Materials. — Besides  the  calculations  to  which 
we  have  alluded,  the  lens  designer  must  also  take 
account  of  the  various  kinds  of  glass  which  he  can 
obtain,  for  by  the  use  of  suitable  glasses  many  results 
can  be  attained  which  were  impossible  by  the  choice 
of  curvatures  alone. 

Of  late  years  Schott  of  Jena  has  succeeded  in  pro- 
ducing a series  of  glasses,  giving  a very  wide  range  of 
properties,  and  it  has  in  consequence  become  possible  to 
construct  lenses  which  can  be  used  with  a much  larger 
aperture  than  formerly. 

Dr.  Paul  Rudolph  of  Jena  has  published  a very 
interesting  fact  about  Jena  glass. 

In  an  achromatic  combination  made  of  two  lenses, 
it  is  necessary,  to  secure  convergence,  that  the  convergent 
lens  should  have  the  least  dispersive  power;  for  in 
I 84  we  found  that  are  the  focal  lengths  of  the 

component  lenses,  co,  co^  their  dispersive  powers,  and  F 
the  focal  length  of  the  combination. 


1 1 


^ + *^=0  (a)  (6) 


A 


F /i  A 


196 


PHOTOGRAPHIC  OPTICS 


From  {a)  we  see  that  since  the  dispersive  powers  are 
positive,  /j  and  must  be  of  opposite  signs,  or  one  lens 
convergent,  the  other  divergent  ; let  represent  the 
convergent  lens,  it  will  then  be  negative.  If  the  system 
is  to  be  convergent  F must  be  negative,  and  hence  we 
must  have  numerically ; and  hence  from  {a) 

> 602  or  the  convergent  lens  has  the  least  dispersive 
power. 

On  inspection  of  the  tables  in  § 81  it  will  be  seen 
that  the  refractive  index  and  the  dispersive  power 
increase  together,  for  if  we  calculate  the  dispersive 
powers  for  the  lines  B,  H for  the  glasses  mentioned  in 
these  tables  we  get 


Refracting  Medium. 

M-I. 

B,  H. 

Dispersive  Power. 

Crown  Glass 

13 

•5243 

•0204 

•0390 

Ditto 

9 

•5258 

•0207 

•0394 

Ditto 

U 

•5548 

•0246 

•0443 

Flint  Glass 

3 

•6020 

•0384 

•0638 

Ditto 

30 

•6236 

•0424 

•0680 

Ditto 

23 

•6266 

•0431 

•0688 

Ditto 

13 

•6277 

•0433  i 

•0690 

where  the  quantity  given  in  the  B,  H column  is 
the  difference  between  the  refractive  indices  for  the 
lines  B and  H. 

Hence  the  convergent  lens  when  made  with  the  old 
materials  must  have  not  only  the  lesser  dispersive 
power,  but  also  the  lesser  refractive  index. 

But  the  most  favourable  arrangement  for  correcting 
astigmatism  in  a doublet,  is  that  in  one  of  the  two 
elements  the  convergent  lens  should  have  the  greater 
index  of  refraction ; this  was  impossible,  as  we  have 
seen,  with  the  old  glasses,  but  has  been  rendered  possible 
by  the  use  of  Jena  glass,  in  which  the  dispersive  power 
does  not  necessarily  increase  with  the  index  of  refraction. 
95.  Flare  Spot. — This  defect  consists  of  a bright 


THE  CORRECTION  OF  ABERRATION 


197 


patch  of  light  in  the  centre  of  the  field,  and  it  is  due  to 
reflections  at  the  surfaces. 

It  is  a fact,  well  known  by  experiment,  that  when  a 
ray  of  light  passes  from  one  medium  to  another,  both 
reflection  and  refraction  take  place,  the  quantity  of 
light  reflected  being  as  a rule  greater  the  larger  the 
angle  of  incidence. 

Since  the  incident  pencil  is  limited  by  the  diaphragm 
the  flare  spot  may  in  some  sense  be  regarded  as  the 
image  of  the  diaphragm  ; an  example  of  this  is  shown  in 
Fig.  62.  Here  C D is  the  aperture  in  the  diaphragm  ; 
the  incident  rays  being  parallel  converge  to  the  principal 
focus  F,  and  the  plate  E H cuts  the  axis  at  this  point ; 
the  reflected  rays  are  shown  by  dotted  lines,  these,  after 
two  reflections  inside  the  lens  are  refracted  out,  giving 
a cone  the  section  of  which  by  the  plate  is  a circle  with 
A B as  diameter.  The  patch  represented  by  A B is 
therefore  the  flare  spot. 

The  figure  represents  the  case  of  a single  lens,  but 
since  with  a combination  there  are  more  reflecting 
surfaces  the  danger  of  a flare  spot  will  be  greater,  and 
there  may  be  more  than  one  such  spot. 

The  intensity  of  illumination  of  a flare  spot  can  never 
be  very  great,  because  it  is  formed  by  two  reflections  at 
least,  at  each  of  which  a great  deal  of  the  light  is 
refracted  ; still  it  may  be  enough,  specially  if  the  spot  be 
small,  to  spoil  the  picture. 

96.  Correction  of  the  Flare  Spot. — In  the  case  of  the 
thin  lens  (Fig.  62),  it  is  obviously  of  no  use  to  move 
the  diaphragm,  the  incident  rays  being  parallel ; but  in 
a compound  lens,  where  the  diaphragm  is  between  the 
two  elements,  some  alteration  can  be  effected  by  moving 
it,  for  the  rays  incident  on  the  second  lens  are  not 
parallel  to  the  axis. 

But  when  a lens  exhibits  a bad  flare  spot,  very  little 
can  be  done  to  get  rid  of  it,  and  the  design  must  be 
reconsidered. 

Since  reflections  always  take  place  inside  the  lens,  the 


198 


PHOTOGRAPHIC  OPTICS 


formation  of  a flare  spot  cannot  be  avoided,  but  it  may 
be  possible  so  to  construct  the  lens,  that  the  spot  may 


THE  COERECTION  OF  ABERRATION 


199 


Fig.  63. 


be  very  large.  The  effect  of  this  is  two  fold  : — Firstly,  the 
flare  spot  covers  the  whole  plate  so  that  all  parts  are 


200 


PHOTOGRAPHIC  OPTICS 


affected  equally,  and  secondly,  since  the  light  is  spread 
over  a large  area,  its  intensity  is  much  diminished.  An 
example  is  given  in  Fig.  63,  which  corresponds  to 
Fig.  62  ; the  flare  spot  is  here  so  large  that  it  cannot 
be  indicated  in  the  figure. 

97.  Correction  of  Spherical  Aberration  and  Astig- 
matism by  means  of  a Diaphragm. — The  aberration 
and  astigmatism  of  a lens  already  constructed  may  be 
lessened  by  means  of  a diaphragm.  It  has  been  shown 
that  as  long  as  the  greatest  breadth  of  the  section  of  a 
pencil  of  light  by  the  plate  does  not  exceed  a length 
which  we  have  called  26,  the  resulting  patch  of  light 
will  appear  to  be  a point.  The  section  of  such  a pencil 
if  anywhere  too  broad  can  obviously  be  reduced  by 
reducing  the  size  of  the  incident  pencils  by  a diaphragm. 
The  proper  position  of  the  diaphragm  needs  some  con- 
sideration ; in  most  cases  this  can  be  found  only  by 
experiment  or  calculation,  but  a few  general  remarks 
may  be  made. 

When  the  lens  is  to  be  used  with  a very  small  angle, 
the  diaphragm  may  be  put  close  to  the  lens,  which  has 
the  effect  of  making  all  but  the  central  portion  of  the 
lens  ineffective  ; but  if  the  lens  have  a fairly  large 
angle,  a diaphragm  close  to  it  will  give  very  bad 
definition  for  pencils  at  all  oblique.  This  is  to  be 
expected,  for  we  have  remarked  (§74),  that  the  greater 
the  inclination  of  the  incident  ray  to  the  normal  to  the 
surface,  the  greater  will  be  the  astigmatism.  It  will 
therefore  be  of  advantage  to  place  the  diaphragm  at  a 
little  distance  from  the  lens,  for  a little  examination 
will  show  that  the  angle  of  incidence  of  the  pencil  at 
the  first  surface  is  always  less  than  in  the  former 
case. 

In  the  particular  case  of  a meniscus  lens,  the  concave 
surface  should  be  turned  to  receive  the  incident  light, 
and  the  diaphragm  should  be  placed  at  the  centre  of 
curvature  of  the  surface  (Fig.  64)  ; the  effect  of  this  is 
that  all  the  incident  pencils  strike  the  first  surface 


THE  COERECTION  OF  ABERRATION 


201 


normally,  and  since  the  pencils  are  small,  very  little 
aberration  is  caused  by  the  first  refraction. 


202 


PHOTOGRAPHIC  OPTICS 


98.  Depth  of  Focus.  — When  a diaphragm  is  used  to 
reduce  the  size  of  pencils,  if  the  most  oblique  pencils 
are  small  enough,  those  less  oblique  will  be  smaller  than 
they  need  be.  If  any  one  of  these  latter  pencils  fall 
on  a screen  which  is  moved  about,  there  will  be  a 
considerable  length  in  which  the  greatest  breadth  of 
the  cross  section  is  less  than  2e,  the  greatest  breadth 
permissible  for  definition. 

Imagine  now  that  along  every  secondary  axis  are 
marked  off  the  extreme  points  at  which  the  greatest 
breadth  of  the  section  is  less  than  26 ; if  this  is  done 
for  all  the  pencils  the  two  series  of  points  will  lie  on  two 
surfaces,  which  will  enclose  a volume  at  all  points  of 
which  the  definition  will  be  good  enough. 

Hitherto  we  have  practically  assumed  that  to  get 
a good  picture  the  circles  of  least  confusion  of  all  the 
small  pencils  must  lie  on  the  plate,  but  now  we  see 
that  if  the  plate  lie  within  the  focal  volume,  the  picture 
will  appear  sharp  all  over,  even  if  the  surface  containing 
the  circles  of  least  confusion  is  not  nearly  a plane. 
Besides  this,  since  the  picture  is  sharp  as  long  as  the 
plate  is  within  the  focal  volume,  we  shall,  if  the  plate 
is  well  inside  this  volume,  be  able  to  move  it  about 
inside  it,  and  thus  have  a certain  latitude  of  position  in 
focussing  ; or  in  other  words,  the  use  of  the  diaphragm 
gives  depths  of  focus. 

In  Fig.  65,  Nj,  N2  are  the  nodal  points  (the  lens 
not  being  shown),  CAD  and  C B D are  the  sections 
of  the  surfaces  on  which  lie  the  extreme  positions  on 
the  secondary  axis  as  defined  above,  the  included  area  is 
the  section  of  the  focal  volmm'e  ; C F D is  the  section 
of  the  surface  on  which  lie  all  the  circles  of  least 
confusion. 

If  a plate  occupy  the  position  indicated  by  the 
dotted  line,  and  its  diagonal  is  not  greater  than  P L,  it 
will  lie  entirely  within  the  focal  volume,  and  the  picture 
will  be  sharp  all  over  it.  If  the  diagonal  of  the  plate 
be  shorter  than  P L,  the  plate  can  be  moved  slightly 


THE  CORRECTION  OF  ABERRATION 


203 


backwards  and  forwards  without  any  portion  emerging 
from  the  focal  volume. 

99.  Correction  of  Distortion. — In  § 77  it  has  been 


204 


PHOTOGRAPHIC  OPTICS 


explained  at  length  that  the  use  of  a diaphragm  gives 
rise  to  distortion ; if  the  diaphragm  be  placed  in  front 
of  the  lens,  straight  lines  near  the  edge  of  the  portion  of 
the  object  are  represented  by  curved  lines,  which  are 
concave  towards  the  axis ; if  the  diaphragm  be  placed 
behind  the  lens,  the  lines  are  curved  and  convex 
towards  the  axis. 

To  correct  thie  defect  two  or  more  groups  of  lenses 
are  used,  the  diaphragm  being  placed  between  two  of 
the  groups ; the  result  of  this  is  that  the  diaphragm 
being  behind  the  front  element,  tends  to  make  lines 
convex  to  the  axis,  but  since  it  is  in  front  of  the  other 
element,  it  tends  to  make  them  concave  to  the  axis. 

Hence,  if  the  diaphragm  be  properly  placed,  the  two 
tendencies  correct  each  other,  and  the  resulting  lines  are 
straight. 

100. — We  have  now  glanced  briefly  at  the  main 
points  to  be  considered  in  correcting  aberration  and  in 
designing  lenses.  It  is  impossible,  within  our  present 
limits,  to  give  any  adequate  idea  of  the  subject ; those 
who  wish  for  more  information  should  consult  M. 
Martin’s  paper,  previously  quoted,  on  “la  determination 
des  courbures  des  objectifs,”  where  there  is  a history  of 
the  development  of  the  subject,  and  many  references  to 
original  papers  ; besides  this,  M.  Martin  gives  the  actual 
work  in  a particular  case. 

The  chapter  in  Wallon’s  L^Ohjectif  Photographique 
on  the  correction  of  aberrations  may  also  be  read  with 
advantage. 


CHAPTER  V 


LENS  TESTING 

101.  — The  examination  of  a lens  falls  naturally  into 
two  divisions  : first,  the  determination  of  what  may  be 
called  the  constants  of  the  lens ; and  secondly,  testing 
for  faults  of  workmanship  and  insufficient  correction  of 
the  aberrations. 

The  most  complete  system  of  testing  is  that  devised 
by  Moessard,^  who  has  invented  for  this  purpose  an 
instrument  called  the  tourniquet ; but  within  the  last 
two  or  three  years  a system  of  lens  testing  has  been 
established  at  Kew  Observatory,  the  apparatus  employed 
being  a modification  of  the  tourniquet.  In  Moessard’s 
system  no  account  was  taken  of  the  expense  or  of  the 
time  required  to  make  the  tests,  the  object  being  to 
examine  a lens  completely  irrespective  of  other  consider- 
ations, while,  on  the  other  hand,  the  object  of  the  Kew 
system  is  to  provide  a really  useful  though  not  elaborate 
test  for  a reasonable  charge. 

Both  systems  will  be  described  in  turn,  but  first  we 
shall  give  several  methods  for  finding  the  most  impor- 
tant constant  of  a lens,  the  principal  focal  length. 

102.  Measurement  of  Principal  Focal  Length. — We 
have  defined  the  principal  focal  length  as  the  distance 
between  the  principal  focus  of  the  lens  and  the  nodal 
point  of  emergence ; but  this  is  not  always  what  is  given 

^ Etiid^.  des  Lentilles  ct  Ohjcctifs  Photograpliiqucs^  par  P.  Moessard. 
Gauthier- Villars,  Paris,  1889. 


205 


206 


PHOTOGRAPHIC  OPTICS 


by  the  makers.  Sometimes  the  distance  from  the  prin- 
cipal focus  to  the  diaphragm  is  given,  sometimes  the 
distance  from  the  principal  focus  to  the  back  surface  of 
the  lens,  called  the  hack  focus. 

For  some  purposes  the  distance  between  the  principal 
focus  and  the  diaphragm  is  near  enough  to  the  true 
focal  length,  but  it  will  not  do  for  calculations  in  con- 
nection with  enlargements  or  reductions  where  accuracy 
is  required,  and  for  most  purposes  the  back  focus  is 
quite  misleading. 

In  the  case  of  single  lenses,  which  may  be  regarded 
as  thin,  and  of  symmetrical  combinations,  which  may  be 
regarded  as  equivalent  to  thin  lenses  placed  at  the 
diaphragm,  either  of  the  two  following  methods  may  be 
adopted. 

(1)  Focus  a distant  object  on  the  ground  glass,  and 
then  measure  the  distance  between  the  ground  glass  and 
the  lens,  or  in  the  case  of  a combination  between  the 
ground  glass  and  the  diaphragm ; this  is  the  principal 
length. 

(2)  Place  upright  in  the  front  of  the  lens  a foot  rule 
or  other  divided  scale,  then  by  trial  place  and  adjust 
the  camera  so  that  the  image  of  the  scale  on  the  ground 
glass  is  of  the  same  size  as  the  object,  which  can  be 
tested  by  measuring  with  a scale  similar  to  that  used 
as  object.  [It  is  of  course  not  meant  that  the  image  of 
the  whole  of  the  scale  must  be  got  on  the  ground  glass, 
but  that  the  sizes  of  the  divisions  in  the  object  and  image 
should  be  equal.]  The  ground  glass  and  scale  are  now 
conjugate  foci,  and  since  the  sizes  of  the  object  and 
image  are  equal,  they  must  be  at  equal  distances  from 
the  centre  of  the  lens. 

The  relation  connecting  u and  the  distances  of 
object  and  image  from  the  centre  of  the  lens,  we  know 
to  be 


1 __  1 _ 1 
u f 


LENS  TESTING 


207 


and  the  object  and  image  being  at  equal  distances  on 
opposite  sides  of  the  lens, 

V — — u 

hence  we  get  from  the  previous  relation 
— =1//  or  /=  — ul2 

but  the  distance  apart  of  the  ground  glass  and  scale  is 
2u,  hence  the  focal  length  is  one  quarter  of  the  distance 
between  the  ground  glass  and  the  scale  when  the  object 
and  image  are  equal. 

In  performing  the  experiment,  it  should  be  remem- 
bered that  the  object  must  be  distant  from  the  lens 
twice  its  focal  length,  or  no  sharp  image  can  be  pro- 
duced ; trouble  and  loss  of  time  are  often  caused  by 
placing  the  object  too  near  the  lens  to  begin  with. 

In  many  cases,  the  extension  of  the  camera  is  not 
enough  to  allow  the  method  to  be  adopted  as  described, 
but  it  is  not  hard  to  carry  it  out  without  the  camera. 
Fix  the  lens  in  a suitable  firm  support  which  can  be 
moved  about  on  a table,  also  fix  vertically  in  movable 
supports  two  similar  divided  scales ; place  one  scale 
behind  the  lens,  and  then  with  an  eye-lens  such  as 
watchmakers  use,  look  for  the  image  of  the  scale. 
When  this  image  is  found,  place  the  other  scale  along- 
side of  it,  and  examine  to  see  if  the  divisions  in  the 
image  are  of  the  same  lengths  as  those  on  the  scale.  If 
the  two  sets  of  divisions  are  not  of  the  same  length,  they 
can  be  made  so  by  moving  the  lens  and  scales  about. 
When  this  adjustment  has  been  made,  remove  the  lens 
and  measure  the  distance  between  the  scales ; one 
quarter  of  this  is  the  focal  length. 

This  method  may  seem  harder  than  the  former,  in 
which  the  camera  was  employed,  but  it  does  not  prove 
so  in  practice,  and  it  is  more  satisfactory,  for  it  is  easier 
to  measure  the  distance  between  two  scales  than  between 
a scale  and  the  ground  glass. 

(3)  Another  method  which  has  the  advantage  of 
giving  the  true  focal  length,  measured  from  the  nodal 


208 


PHOTOGRAPHIC  OPTICS 


point,  is  to  fix  a scale  in  front  of  the  camera  as  before, 
and  to  arrange  it  so  that  the  image  on  the  ground  glass 
is  of  the  same  size  as  the  object,  then  move  the  ground 
glass  to  focus  up  sharply  some  distant  object ; the  focal 
length  required  is  equal  to  the  distance  through  which  the 
ground  glass  has  been  moved. 

For,  in  the  first  case,  when  object  and  image  were 
equal,  the  ground  glass  was  distant  twice  the  focal  length 


A 


Fig.  66. 


from  the  nodal  point ; and,  in  the  second  case,  when  the 
distant  object  was  in  focus,  it  was  at  the  principal 
focus. 

(4)  The  following  method,  due  to  Grubb,  is  quoted 
from  Monkhoven’s  Photographic  Optics  : — 

“Let  A B,  Fig.  66,  be  objects  widely  separated 
situated  on  the  horizon ; C the  objective,  screwed  on  to 
a camera  placed  on  a well-levelled  table.  On  bringing 
them  to  a focus  on  the  ground  glass,  we  find  that  the 


LENS  TESTING 


209 


objects  D and  E form  the  limits  of  the  image  on  the 
ground  glass.  D C E is  the  angle  included  by  the  lens. 
Draw  on  the  middle  of  the  ground  glass  a vertical  right 
line,  and  turn  the  camera  until  the  point  E falls  on  this 
line. ' With  a pencil  pressed  against  the  side  of  the  camera 
draw  the  right  line  c e.  Turn  the  camera  towards  the 
point  D until  this  point  falls  on  the  line  traced  on  the 
ground  glass.  Draw  the  right  line  c c?  in  the  same  way 
as  c e was  done.  If  this  line  does  not  cut  c e,  prolong 
it  until  it  does.  It  is  clear  that  the  angle  e c d equal 
to  D C E.  Therefore,  by  placing  the  centre  of  a pro- 
tractor at  c,  the  number  of  degrees,  e d,  is  read  off  ; 
that  is,  the  angle  included  by  the  lens. 

“ Its  absolute  focal  length  is  thus  obtained  : — 

‘‘  Measure  on  the  ground  glass  the  distance  of  the 
points  with  a compass,  and  take  half  of  them.  Bisect 
the  angle  e c by  a straight  line  c against  which  place 
a square  rule.  Carry  the  half-distance,  D E (measured 
on  the  ground  glass),  on  the  square  rule,  and  make  f g 
equal  to  this  half-distance,  and  perpendicular  to  c f 
Then  fc  will  be  the  true  focal  length  of  the  objective, 
which,  when  once  known,  permits  the  size  of  images  to 
be  calculated.’’ 

(5)  Several  other  methods  are  given  in  Wallon’s 
L’ Ohjectif  Photographique^  pp.  129 — 140,  ed.  1891. 


I.  M.  Moessard’s  System  and  the  Tourniquet. 

103.  Desiderata. — Thequantities  determined  and  tests 
made  are  as  follows  : — 

(1)  The  principal  focal  length  and  positions  of  nodal 
points. 

(2)  The  form  of  the  principal  focal  surface. 

(3)  The  depth  of  focus,  or  the  principal  focal 
volume. 

(4)  Astigmatism. 

(5)  Distortion, 

p 


210 


PHOTOGRAPHIC  OPTICS 


(6)  The  field  of  the  lens. 

(7)  Brightness  and  transparency. 

(8)  Achromatism. 

The  first  five  and  the  last  of  these  quantities  have 
already  been  fully  treated ; the  remaining  two  need  a 
little  explanation.  By  the  field  of  the  lens  is  meant  in 
general  the  angle  of  the  cone  enclosing  the  largest  space 
over  which  the  objective  will  furnish  a sharp  picture ; 
this  cone  has  its  vertex  at  the  nodal  point  of  emerg- 
ence. 

The  brightness  of  the  image  depends  on  the  trans- 
parency of  the  lens,  which  may  be  defined  to  be  the 
ratio  of  the  quantity  of  light  which  actually  gets  through 
the  lens,  to  the  quantity  which  would  get  through  if 
the  glass  were  removed,  and  the  mounting  and  stop  left 
unaltered ; it  is  an  important  matter,  for  the  brighter 
the  image,  the  shorter  will  be  the  exposure.  All  lenses 
waste  some  of  the  light  which  falls  on  them,  because  of 
the  reflection  and  scattering  at  the  surfaces,  and  some- 
times, if  the  glass  is  not  of  good  quality,  a considerable 
amount  of  light  is  absorbed.  Moessard  divides  his 
operations  into  five  experiments. 

(1)  Determination  of  the  nodal  points  and  principal 
focal  length. 

(2)  Determination  of  the  principal  focal  surfaces, 
depth  of  focus,  astigmatism,  maximum  flat  field. 

(3)  Measurement  of  distortion  and  of  the  field  free 
from  distortion. 

(4)  Measurement  of  transparency  and  field  of  equal 
brightness. 

(5)  Test  for  achromatism  and  determination  of  visual 
and  chemical  foci. 

In  the  course  of  these  five  experiments,  the  faults  of 
construction,  such  as  bad  mounting  and  centering, 
irregularities  in  the  lenses,  etc.,  are  detected. 

104.  Description  of  the  Tourniquet. — This  apparatus 
resembles  an  ordinary  camera  in  appearance  ; it  consists 
(Fig.  67)  of  a carrier  which  can  be  worked  with 


LENS  TESTING 


211 


rack  and  pinion,  and  is  connected  to  a cubical  box  in 
front  by  bellows.  On  the  carrier  is  a small  ground 


Fig.  67. 


glass  e in  a frame  hinged  at  the  side  so  that  it  can 
be  thrown  back  ; there  is  also  hinged  to  it  and  opening 


212 


PHOTOGRAPHIC  OPTICS 


on  the  opposite  side,  a panel,  carrying  at  its  centre  a 
micrometer  scale  m,  divided  to  tenths  of  a millimetre, 
and  furnished  with  a simple  microscope  to  read  it.  Thus 
when  the  ground  glass  is  folded  back,  and  the  micro- 
meter is  in  place,  the  image  can  be  viewed  by  the 
microscope,  and  its  size  accurately  measured  against  the 
scale;  the  scale  can  be  twisted  through  an  angle  of  90°, 
so  that  lines  in  all  directions  can  be  measured. 

The  cubical  box  in  front  has  two  small  circular  open- 
ings ; one,  c,  in  front,  the  other  into  the  bellows  ; the 
front  of  the  box  can  open,  as  shown  in  the  figure,  being 
hinged  at  the  side.  Inside  the  cubical  box  is  a smaller 
one,  open  at  two  sides,  which  can  turn  about  a vertical 
axis  R R',  and  the  dimensions  are  so  chosen  that  it  can 
turn  right  round.  The  axis  R R'  projects  through  the 
top  of  the  outer  fixed  box,  and  can  be  moved  from  the 
outside  by  the  metal  arm  M M'.  Inside  the  movable 
box  is  a vertical  panel,  movable  forward  and  backward 
by  the  screw  Y Y'  ; to  the  centre  of  this  panel  is  fixed 
the  lens  to  be  tested.  To  enable  lenses  of  all  sizes  to  be 
tested,  several  panels  with  various  flanges  are  kept ; 
also  the  panel  is  not  quite  as  broad  as  the  box,  allow- 
ing some  side-play,  which  is  useful  to  centre  the  lens 
correctly. 

The  metal  arm  M M'  carries  at  its  ends  sights  by 
means  of  which  it  can  be  directed  to  any  object 
required,  and  it  is  pierced  at  g g by  two  holes  into 
which  fit  pins  by  which  it  can  be  fixed  in  the  zero 
position  parallel  to  the  length  of  the  apparatus.  On 
the  top  of  the  box  is  fixed  a circular  arc  with  holes  at 
equal  angular  distances,  into  which  the  pins  through 
g can  fit,  which  enables  the  lens  to  be  set  with  its 
axis  making  various  known  angles  with  the  axis  of  the 
tourniquet. 

Lastly  the  base  board  along  which  the  carrier  slides 
is  fitted  with  a scale  and  vernier  to  measure  the  distances 
through  which  it  moves  ; the  zero  of  the  scale  is  at  the 
centre  of  the  axis  R R', 


LENS  TESTING 


213 


The  whole  apparatus  is  supported  on  a substantial 
tripod. 

To  centre  the  lens  and  make  its  axis  pass  exactly 
through  the  axis  of  rotation,  the  tool  shown  in  Fig. 
68  IS  used  ; this  consists  of  a tube  1 1'  which  fits  closely 
the  axis  R,  which  is  hollow  ; at  the  lower  end  is  fixed  a 
rider  with  its  sides  equally  inclined  to  the  vertical. 
This  is  pushed  down  near  the  lens,  and  the  lens  is 
adjusted  till  it  touches  A B and  B C at  the  same  time. 

105.  Experiment  1.  Determination  of  the  Nodal 
Points  and  Principal  Focal  Length. — To  find  the  posi- 
tion of  the  nodal  points  we  must  make 
use  of  the  property  of  the  nodal 
point  of  emergence  proved  in  § 65 — 
i.  e.  if  the  lens  be  rotated  about  an 
axis  through  this  nodal  point  at  right 
angles  to  the  axis  of  the  lens,  the 
picture  on  the  ground  glass  will 
remain  stationary,  provided  the  angle 
of  rotation  be  not  very  large. 

The  picture  of  a distant  object  is 
focussed  on  the  ground  glass,  and  the 
lens  is  then  moved  from  side  to  side 
by  the  arm  MM';  if  the  picture 
moves  the  same  way  as  the  handle,  then  the  nodal 
point  is  between  the  image  and  the  axis,  but  if  it 
moves  the  other  way  the  point  is  beyond  the  axis. 
The  lens  is  then  adjusted  as  required,  by  the  screw 
Y Y',  and  the  test  repeated,  and  so  on  till  the  image 
stays  still.  For  greater  accuracy  the  ground  glass  may 
be  removed  and  the  tests  repeated,  while  the  image  is 
viewed  through  the  microscope  against  the  micrometer 
scale. 

When  the  adjustment  is  complete  the  nodal  point  of 
emergence  lies  on  the  axis  R R',  and  the  focal  length, 
being  the  distance  from  the  axis  to  the  ground  glass, 
is  read  off  on  the  scale  along  the  base  board  of  the 
instrument. 


214 


PHOTOGRAPHIC  OPTICS 


The  position  of  the  nodal  point  is  marked  on 
the  mounting  of  the  objective,  by  passing  down  the 
tube  tt'  ^ tool  which  when  tapped  makes  a V mark  ; 
the  angle  of  the  V coincides  exactly  with  the  axis 
and  is  therefore  the  position  of  the  nodal  point. 

To  find  the  other  nodal  point,  turn  the  lens  right 
round  with  the  inner  box  and  repeat  the  experiment, 
marking  the  nodal  point  as  before ; the  focal  length 
thus  found  should  be  the  same  as  that  in  the  first 
case. 

If  an  examination  of  the  component  lenses  is  required, 
these  can  be  tested  in  a similar  manner  by  screwing 
out  the  lenses  in  turn  from  the  mounting. 

106.  Experiment  2.  Determination  of  the  Principal 
Eocal  Surface,  Depth  of  Focus,  Astigmatism,  Maxi- 
mum Flat  Field. — This  experiment  must  be  performed 
with  every  diaphragm  to  be  tested. 

The  nodal  point  of  emergence  is  placed  on  the  axis 
of  rotation  as  before  and  the  focal  length  obtained  ; the 
lens  is  then  turned  through  a definite  angle  by  the  arm 
M and  fixed  in  that  position.  The  object  is  again 
focussed  and  the  distance  of  the  screen  from  the  nodal 
point  again  noted  ; this  process  is  repeated  at  regular 
angular  intervals,  on  either  side  of  the  mid  position  till 
the  extremity  of  the  sharp  field  of  the  lens  is  reached. 
The  angle  at  which  the  image  ceases  to  be  sharp  is 
noted,  and  also  the  angle  at  which  the  light  is  just  all 
shut  off  by  the  mounting,  giving  the  angles  of  the  cone 
of  sharpness  and  cone  of  illumination  (§  114). 

The  results  of  the  observations  are  plotted  on  a 
diagram  of  which  Fig.  69  is  a reduced  copy;  the  dis- 
tances are  measured  off  from  N,  along  the  lines  corre- 
sponding to  the  angles  through  which  the  lens  is  turned. 
Suppose,  for  instance,  that  the  lens  is  of  5 inches  focal 
length,  if  the  points  found  are  joined  by  a continuous 
line  we  shall  get  a curve  such  as  A F B,  wliich  is  the 
section  of  the  principal  focal  surface  by  the  plane  of  the 
diagram. 


LENS  TESTING 


215 


To  find  how  far  the  field  is  practically  flat,  the 
tangent  at  F is  drawn  to  A F B,  and  the  length  B B' 


is  marked  off  so  that  no  point  on  it  is  distant  more 
than  one-  fiftieth  of  an  inch  from  the  curve  A F B ; 
points  on  B B'  will  then  be  practically  indistinguishable 


PHOTOGRAPHIC  OPTICS 


216 


from  the  curve,  or  the  corresponding  portion  of  the 
curve  is  practically  a straight  lined 

To  test  the  symmetry  of  the  lens,  twist  it  through 
any  required  angle  about  its  axis  and  repeat  the 
measures ; if  the  lens  is  symmetrical  about  the  axis,  the 
values  now  obtained  should  be  identical  with  the 
former. 

Depth  of  Focus. — For  this  measure  a diagram  (Fig.  7 0) 
composed  of  black  and  white  triangles  is  employed  ; it 
is  placed  at  a fair  distance  and  focussed  so  that  the 
breadth  of  the  image  ah  or  h c oi  the  triangles  across 
some  line  ah  c is  less  than  one  two-hundredth  of  an 


Fig.  70. 


inch.  In  default  of  the  diagram  the  lens  may  be 
focussed  on  some  dark  objects,  distant  chimneys  for 
instance,  so  that  the  breadth  of  some  detail  is  one  two- 
hundredth  of  an  inch. 

This  done,  the  ground  glass  is  moved  backwards  and 
forwards  and  the  positions  are  noted  at  which  the  points 
a and  h become  indistinguishable  or  the  detail  of  the 
distant  object  disappears ; this  gives  the  depth  of  focus 
along  the  axis  of  the  lens. 

^ In  this  description  and  elsewhere  English  measures  have 
been  substituted  for  French  measures,  the  nearest  convenient 
approximation  being  taken. 


LENS  TESTING 


217 


The  lens  is  then  turned  and  fixed  at  various  inclin- 
ations and  the  observations  repeated. 

The  lengths  so  obtained  are  plotted  on  the  same 
diagram  as  that  on  which  the  focal  surface  was  plotted 
(Fig.  69);  the  continuous  curves  A C B,  A D B drawn 
through  these  points  are  sections  of  the  bounding 
surfaces  of  the  principal  focal  volume.  To  find  the 
largest  possible  plane  picture,  draw  within  the  area 
bounded  by  the  curves  the  longest  possible  straight  line 
perpendicular  to  the  axis  ; the  line  will  clearly  be  E D G 
touching  the  curve  A D B at  D.  The  length  of  the 
line  will  be  the  length  of  the  diagonal  of  the  largest 
plate  which  can  be  covered  properly  by  the  lens  with 
the  diaphragm  used  ; the  experiment  can  if  required  be 
performed  with  other  diaphragms. 

Astigmatism. — To  determine  this  use  a fairly  distant 
object  marked  with  horizontal  and  vertical  lines. 
Place  the  lens  at  the  angle  at  which  it  is  required  to 
determine  the  astigmatism.  Move  the  ground  glass 
about  till  in  one  position  the  horizontal  lines  are  sharp 
and  the  vertical  ones  blurred,  and  in  the  other  the 
vertical  lines  are  sharp  while  the  horizontal  ones  are 
blurred.  In  these  positions  the  ground  glass  receives 
the  horizontal  and  vertical  focal  lines  of  the  pencil ; 
the  distance  between  the  two  positions  of  the  ground 
glass  is  therefore  the  distance  between  the  focal  lines, 
and  this  we  have  taken  as  the  measure  of  the  astigma- 
tism. 

107.  Experiment  3.  Measurement  of  Distortion,  and 
of  the  Field  free  from  Distortion. — This  measure  de- 
pends on  the  fact  that  when  a lens  exhibits  distortion,  the 
displacement  of  the  picture  when  the  lens  is  rotated  in 
Experiment  1 is  due  not  only  to  the  nodal  point  not 
being  on  the  axis,  but  also  to  the  distortion ; for  sup- 
pose, for  example,  that  the  nodal  point  of  emergence  is 
by  some  means  placed  on  the  axis  of  rotation,  then, 
according  to  the  elementary  theory,  the  picture  should 
not  move  as  the  lens  is  rotated,  which  arises  from  the 


218 


PHOTOGRAPHIC  OPTICS 


property  that  the  lines  joining  the  corresponding  points 
of  object  and  image  to  the  corresponding  nodal 
points  are  parallel.  But  we  have  seen  (§  78)  that  the 
distortion  arises  from  the  displacement  of  the  image, 
by  aberration  from  the  place  assigned  to  it  by  the 
elementary  theory.  A little  consideration  will  then 
show  that  even  if  the  nodal  point  of  emergence  is  on 
the  axis  of  rotation,  the  picture  will  at  points  not  on 
the  axis  be  displaced  by  the  rotation ; this  will  in 
most  cases  be  too  small  to  detect  (at  any  rate  near  the 
centre  of  the  picture  if  the  rotation  is  small),  but  will 
become  noticeable  if  the  angle  is  large. 

In  most  cases  then  the  nodal  point  can  be  placed  on 
the  axis  as  in  Experiment  1,  by  moving  the  lens 
through  a small  angle  ; when  this  adjustment  is  made, 
the  lens  should  be  displaced  through  a large  angle,  and 
the  distance  noted  through  which  the  point  in  the  pic- 
ture, originally  in  the  centre  of  the  field,  is  displaced 
along  the  micrometer  scale. 

The  length  thus  found  is  the  distortion,  with  the 
stop  used,  for  a pencil  making  an  angle  with  the  axis 
equal  to  that  through  which  the  lens  was  displaced. 

The  procedure  necessary,  if  the  distortion  is  large 
enough  to  interfere  with  the  first  adjustment,  is 
explained  by  Mbessard  in  his  book  ; it  would  occupy 
too  much  space  to  give  it  here. 

The  extent  of  the  field  free  from  distortion  is  found 
by  observing  the  angle  through  which  the  lens  can  be 
turned  without  the  central  point  of  the  picture  moving 
more  than  one  two-hundredth  of  an  inch. 

108.  Experiment  4.  Measurement  of  Transparency 
and  Field  of  Equal  Brightness. — Moessard  has  given  a 
method  of  estimating  the  transparency  of  a lens,  but  it 
has  the  disadvantage  of  giving  it  only  for  visual  and 
not  for  actinic  rays ; this  is  practically  useless  for 
photographic  purposes,  as  the  actinic  rays  are  much  cut 
off  by  a yellow  tinge  in  the  glass  which  produces  very 
little  visual  effect. 


LENS  TESTING 


2J9 


We  shall  therefore  omit  the  account  of  this  experi- 
ment, referring  those  who  wish  to  read  of  it  to  Moes- 
sard’s  or  Wallon’s  book.  In  practice  the  comparison 
of  two  lenses  is  best  made  photographically  ; a method 
for  this  is  described  in  § 125. 

The  field  of  equal  brightness  is  considered  among  the 
Kew  tests,  § 115,  NTo.  16. 

109.  Experiment  5.  Test  for  Achromatism  and 
Determination  of  Visual  and  Chemical  Foci. — In  a sheet 
of  cardboard  a rectangular  slit  is  made,  about  2 inches 
long  and  half-an-inch  broad,  across  which  are  fastened 
two  threads  at  right  angles,  along  and  across  the  slit. 
The  card  is  placed  in  a window  8 or  10  feet  distant 
from  the  tourniquet,  so  as  to  be  projected  against  the 
sky  or  a white  wall,  and  the  threads  are  carefully 
focussed  with  the  micrometer. 

The  micrometer  is  then  replaced  by  a direct  vision 
spectroscope,  so  placed  that  the  image  found  is  at  the 
distance  of  distinct  vision.  The  image  of  the  slit  seen 
in  the  spectroscope  becomes  a band  of  colour,  composed 
of  the  rays  of  the  solar  spectrum  ; if  the  lens  is  well 
achromatized,  the  edges  of  the  slit  will  be  quite  sharp 
from  one  end  to  the  other. 

If  the  lens  is  not  achromatic  the  positions  of  the  foci 
for  different  colours  can  be  found  by  moving  the 
spectroscope,  and  observing  the  positions  in  which  the 
various  lines  become  sharp. 

This  method,  though  theoretically  satisfactory,  is  not 
practically  convenient,  as  it  requires  the  use  of  a direct 
vision  spectroscope,  which  few  photographers  possess  ; 
the  following  method  is  more  convenient. 

110.  Photographic  Test  of  Achromatism. — The 
simplest  way  to  test  the  achromatism  of  a lens,  or  in 
other  words  the  coincidence  of  the  foci  of  the  visual 
and  chemical  rays,  is  to  photograph  numbered  slips  of 
cardboard  placed  at  slightly  different  distances  from 
the  lens. 

A strip  of  wood  is  taken  (a  convenient  size  is  f inch 


220 


PHOTOGRAPHIC  OPTICS 


square  by  1|-  inches  in  length),  on  this  transverse  cuts 
are  made  at  distances  of  ^ inch — seven  will  be  enough  ; 
seven  thin  strips  of  card  about  1 inch  long  by  f inch 
broad  are  numbered  at  their  extremities  from  1 to  7. 
These  strips  of  card  are  then  stuck  into  the  slits  in  the 
wood,  being  arranged  fan-wise  (Fig.  71),  so  that  when 
viewed  from  the  front  all  the  numbers  at  the  extremities 
are  visible. 


Fig.  71. 


The  strip  of  wood  is  then  placed  one  or  two 
yards  in  front  of  the  camera,  with  its  length  along 
the  axis  of  the  lens  ; the  middle  number  4 of  the 
series  is  clearly  focussed  and  a photograph  taken.  The 
negative  when  developed  is  examined  to  see  which 
number  has  come  out  the  sharpest ; if  4 is  the  sharpest, 
then  the  visual  and  chemical  foci  coincide,  but  if  they 
do  not  it  will  show  their  relative  positions. 

We  can  estimate  the  difference  between  the  princi- 
pal focal  lengths  for  the  two  kinds  of  rays  as  follows. 


LENS  TESTING 


221 


Let  /y  be  the  focal  length  for  the  visual  rays,  f 
that  for  the  chemical  rays,  also  let  u and  v be  the 
distances  of  object  and  image  for  visual  rays,  and  u + 
X and  V similar  quantities  for  the  chemical  rays  ; here 
X and  a are  small  quantities  compared  with  v and  f. 
We  have  then — 

1 _ 1 _ 1 1 1_  _ 1 

V u f ^ V u X / a 

Subtracting  we  get 

1 1 1 1 X ^ — ct 

20  20  X y + f + ^)  y (/  + 

. (y*  “b  — — a 2jj  (u  “b 

or  f'^x-{-xaf=  —w^a-\-axu 

But  a and  x are  small,  hence  we  may  neglect  the 
product  xa^  compared  with  either  x or  a,  and  the  rela- 
tion becomes 

f ^ 

i X = — a or  a — — — ^ X, 

Hence,  if  we  know  the  visual  focal  length,  the  dis- 
tance from  the  lens  of  the  card  which  appears  clearly 
focussed,  and  x the  distance  from  this  card  of  that 
which  photographs  most  clearly,  then  we  can  calculate 
the  difference  between  the  visual  and  chemical  focal 
lengths. 

Exmu'ple. — The  card  numbered  4,  which  is  focussed 
sharply,  is  2 feet  from  the  lens,  /.u  — 2 feet,  the  visual 
focal  length  is  6 inches,  orf  — — 1/2  foot  (convex  lens)  ; 
it  is  found  that  the  card  marked  2 is  sharpest  in  the 
photograph  when  4 is  sharpest  on  the  ground  glass. 
Find  the  difference  between  the  visual  and  chemical 
focal  lengths. 

x = ^ 1/2  inch  = — 1/24  foot,  the  cards  being  1/4 
inch  apart. 

^ Readers  acquainted  with  the  differential  calculus  will  easily 
get  this  result  by  differentiation. 


222 


PHOTOGRAPHIC  OPTICS 


Hence 


a 


1 1 

= 7 X 7 X 
4 4 


1 _ 1 

24  ~ 384 


feet  = 


1 

32 


inch. 


Hence  the  chemical  focal  length  in  inches  is 

/+  a = - 6 + 1/32  = - (6  - 1/32) 

or  the  chemical  focal  length  is  numerically  less  than 
the  visual  one  by  one  thirty- second  of  an  inch. 

If  we  put  the  relation  found  above  in  the  form 


we  see  that  the  larger  is  u the  distance  of  the  object 
from  the  lens,  the  larger  will  be  x the  distance  between 
the  cards  for  a given  value  of  a,  the  difference  between 
the  focal  lengths. 

Hence  the  further  the  numbered  cards  are  placed 
away  from  the  lens  the  more  sensitive  will  be  the 
method. 

In  practice  it  is  usually  required  only  to  test  the 
achromatism  of  a lens,  and  one  which  is  not  achromatic 
would  be  rejected  for  ordinary  work.  But  in  scientific 
experiments  it  is  sometimes  necessary  to  use  a single 
uncorrected  lens  for  taking  a series  of  photographs  of 
objects  at  a fixed  distance ; if  so  the  method  given  is 
useful  for  finding  the  proper  relative  distances  of  object 
and  image. 

111.  Examination  of  the  Faults  of  Construction. — • 

The  faults  which  should  be  looked  for  are  in  the 
centering  of  the  lenses  and  the  working  of  the 
surfaces. 

Centering. — The  tourniquet  supplies  a method  for 
testing  the  centering  of  the  component  lenses  of  an 
objective,  for  if  they  are  not  concentric  with  their 
mounting,  the  nodal  points  will  not  lie  on  the  axis  of  the 
mounting  ; it  will  not  then  be  in  general  possible  to 
put  the  nodal  points  on  the  axis  of  rotation,  and  this 
makes  it  impossible  to  make  the  picture  stationary,  as 


LENS  TESTING 


223 


in  Experiment  1.  But  there  are  evidently  two  posi- 
tions in  which  the  nodal  point  is  either  immediately 
above  or  below  the  axis  of  the  mounting,  into  which  the 
lens  can  be  twisted  in  which  the  nodal  point  examined 
is  on  the  axis  of  rotation. 

The  method  of  conducting  the  test  is  to  fix  the  lens 
in  the  tourniquet,  and  to  proceed  to  find  the  nodal 
point  as  already  explained.  If  the  picture  cannot  be 
rendered  stationary,  then  the  centering  is  bad,  but  if 
the  picture  can  be  rendered  stationary,  twist  the  lens 
through  a right  angle,  and  repeat  the  test.  If  the 
picture  is  still  stationary  the  centering  is  good,  if  not 
it  is  bad. 

The  Working  of  the  Surfaces, — If  the  centering  is  good 
the  correctness  of  the  surfaces  may  be  tested  by  finding 
the  position  of  the  nodal  points;  twist  the  lens  through 
an  angle  and  find  them  again,  and  so  on  until  two  or 
three  sets  are  found  ; mark  these  as  before  on  the 
mounting.  If  the  surfaces  are  all  symmetrical  about 
the  axis,  the  marks  so  made  will  lie  on  a circle  whose 
plane  is  perpendicular  to  the  axis  of  the  lens. 

This  test  will  also  show  if  there  is  any  great  want 
of  homogeneity  in  the  material  of  the  lenses. 

Faults  in  the  Glass. — By  this  are  meant  defects  in  the 
glass  itself,  such  as  stride,  veins,  and  colour ; they 
can  be  detected  by  examining  the  lens  in  a good  light, 
turning  it  about  in  all  directions. 


II. — The  Kew  System. 

112. — In  1891-2  the  Kew  Committee  of  the  Boyal 
Society  decided  to  establish  a series  of  tests  for  photo- 
graphic lenses ; a system  was  accordingly  devised  by 
Major  Darwin  and  the  late  superintendent  of  the 
observatory,  Mr.  Whipple. 

As  this  series  of  tests  is  especially  interesting  to 
English  readers,  some  considerable  account  of  it  will  be 


224 


PHOTOGRAPHIC  OPTICS 


given ; the  details  are  quoted  from  a paper  by  Major 
Darwind 

The  object  in  view  is  best  quoted  from  the  paper  : — 

“ The  object  of  the  Committee  was  to  organize  a 
system  by  which  any  one  could  obtain,  on  payment,  an 
impartial  and  authoritative  statement  of  the  quality  of 
a lens  to  be  used  for  ordinary  photographic  purposes, 
and  that  the  fee,  which  had  to  cover  the  cost  of  the 
examination,  should  be  moderate.  This  latter  con- 
sideration acted  as  a serious  restriction,  and  it  was 
consequently  necessary  that  all  the  tests  should  give 
results  of  undoubted  practical  value  to  the  practical 
photographer  ; the  certificate  of  examination  must  be 
recorded  in  the  way  most  generally  useful,  and  in 
language  which  could  not  fail  to  be  understood.  A 
complete  scientific  investigation  of  a lens  from  every 
point  of  view  would  occupy  so  long  a time  as  to  make 
the  necessary  fee  quite  prohibitive,  and,  moreover,  the 
results  would  contain  much  information  which  would 
be  quite  useless  to  the  ordinary  user  of  the  lens. 

“ There  are  undoubted  advantages  in  testing  a lens 
by  the  examination  of  negatives  made  by  it,  but  it  may 
be  here  stated,  once  for  all,  that  the  question  of 
expense  rendered  it  impossible,  for  the  present,  to  adopt 
any  photographic  method  ; eye  observations  alone  have 
to  be  relied  on.” 

In  most  cases  a lens  is  designed  for  a particular 
kind  of  work,  and  for  use  with  a plate  of  a particular 
size,  so  to  shorten  the  examination,  the  person  enter- 
ing the  lens  is  asked  to  state  these  particulars,  and  the 
examination  is  made  for  them  only. 

113. — The  list  of  tests  made  is  best  given  by  quoting 


^ ‘‘  On  the  Method  of  Examination  of  Photographic  Lenses  at 
Kew  Observatory.”  By  Leonard  Darwin,  Major  late  Royal 
Engineers.  Proceedings  of  the  Royal  Society,  No.  318,  December 
1892.  Vol.  52,  p.  403. 

Major  Darwin  has  kindly  given  permission  to  quote  from  the 
paper  and  to  use  the  diagrams. 


LENS  TESTING 


225 


a certificate  of  examination  which  will  afterwards  be 
explained  in  detail ; the  part  in  italics  represents  the 
results  of  the  tests. 


Kew  Obseuvatory,  Richmond,  Surrey. 

Certificate  of  Examination  of*  a Photographic  Lens. 

1.  Number  on  lens,  S876.  Registered  number,  95. 

2.  Description,  landscape  lens.  Diameter,  1'5  inches. 

3.  Maker’s  name,  A.  B. 

4.  Size  of  plate  for  which  the  lens  is  to  be  examined,  6'5  inches 

by  8’5  inches. 

5.  Number  of  reflecting  surfaces, 

6.  Centering  in  mount,  good. 

7.  Visible  defects — such  as  striae,  veins,  feathers,  &c.,  nil. 

8.  Flare  spot,  nil. 

9.  Eflective  aperture  of  stops — 


Number 
engraved  on 
stop. 

Effective 
apertur  ■. 
Inches. 

No.  7-5 

1'32 

No.  10 

1'19 

No.  15 

0'97 

No.  25 

0'75 

No.  50 

O'Jf.9 

No 

No 

//number. 

C.I.  No. 

flS-6 

Ill' 38 

fl9-5 

111-12 

fill -7 

1'35 

fll5-l 

2'26 

fl23 

5 '3 

10.  Angle  of  cone  of  illumination  with  largest  stop  = 68°^ 

giving  a circular  image  on  the  plate  of  ^ 13 '2  inches 
diameter. 

Angle  of  cone  outside  which  the  aperture  begins  to  be 
eclipsed,  with  stop  C.I.  No.  lll’38,  = 20'^,  giving  a cir- 
cular image  on  the  plate  of  4'0  inches  diameter. 

Diagonal  of  the  plate  = 10 '7  inches,  requiring  a field  of 
51\ 

Stop  C.I.  No.  5' 3 is  the  largest  stop  of  which  the  whole 
opening  can  be  seen  from  the  whole  of  the  plate. 

11.  Principal  focal  length,  ^ inches.  Back  focus,  or 

length  from  the  principal  focus  to  the  nearest  point  on 
the  surface  of  the  lenses,  — 10 ‘5  inches. 


^ The  lens  is  focussed  on  a very  distant  object. 

Q 


226 


PHOTOGRAPHIC  OPTICS 


12.  Curvature  of  the  field,  or  of  the  principal  focal  surface. 

After  focussing  ^ the  plate  at  its  centre,  movement 
necessary  to  bring  it  into  focus  for  an  image  1 5 inches 
from  its  centre  = 0’02  inch. 

Ditto  for  an  object  S inches  from  its  centre  = O’OJf.  inch. 

,,  „ =^0'10  „ 

5 „ =0'15  ,, 

13.  Definition  at  the  centre  with  the  largest  stop,  excellent. 

C.I.  stop  No.  1'35  gives  good  definition  over  the  whole  of 
a 6 '5  inch  by  8' 5 inch  plate. 

14.  Distortion.  Deflection  or  sag  in  the  image  of  a straight 

line  which,  if  there  were  no  distortion,  would  run  from 
corner  to  corner  along  the  longest  side  of  a 6'5  inch  by 
8’5  inch  plate  = 0 ’01  inch.^ 

15.  Achromatism.  After  focussing  ^ in  the  centre  of  the  field  in 

white  light,  the  movement  necessary  to  bring  the  plate 
into  focus  in  blue  light  (dominant  wave-length,  4420),  = 
4-  O’OJf-  inch.^  Ditto  in  red  light  (dominant  wave-length, 
6250)  = - O'Ol  inch.^ 

16.  Astigmatism.^  Approximate  diameter  of  disc  of  diffusion^ 

in  the  image  of  a point,  with  C.I.  stop  No. at 

inches  from  the  centre  of  the  plate  = 0 • inch. 

17.  Illumination  of  the  field.  The  figures  indicate  the  relative 

intensity  at  different  parts  of  the  plate.  ^ 


With  C.I.  stop  No.  Ijl’SS. 

At  the  centre 100 

At  3 inches  from  the  centre  67 
At  5’35  ,,  38 


With  stop  No.  O’ 3. 


Ditto  100 

Ditto  82 

Ditto  66 


General  Remarks. — An  excellent  medium  angle  rapid  ohjective, 
practically  free  from  distortion. 

Date  of  issue 


W.  HUGO,  Observer. 

G.  M.  WHIPPLE,  Superintendent. 


^ The  lens  is  focussed  on  a very  distant  object. 

The  sag  or  sagitta  here  given  is  considered  positive  if  the 
curve  is  convex  towards  the  centre  of  the  plate. 

^ Positive  if  movement  towards  the  lens,  negative  if  away  from 
it. 

^ The  lens  is  supposed  to  be  perfect  in  other  respects. 

Note. — The  following  is  the  scale  of  terms  used  : excellent 
good,  fair,  indifferent,  bad. 


“ In  considering  and  in  recording  the  results  of 
examinations,  it  has  been  found  convenient  to  give 
more  exact  meanings  to  certain  expressions  than  have 


LENS  TESTING 


227 


as  yet  been  assigned  to  them.  The  following  definitions 
have  therefore  been  adopted  at  Kew  : — 

“ A narrow  angle  lens  means  one  covering  effectively 
not  more  than  35°. 

A medium  angle  lens  means  one  covering  between 
35°  and  55°. 

“ A wide  angle  lens  means  one  covering  between  55° 
and  75°. 

‘‘  All  extra  wide  angle  lens  means  one  covering  more 
than  75°. 

“ The  CJ.  Afo.  of  a stop  means  the  number  which 
indicates  the  intensity  of  illumination  produced  by  it 
on  the  plate  according  to  the  system  proposed  at  the 
International  Photographic  Congress  of  1889  (see  § 
123). 

“ The  largest  normal  stop  means  the  largest  stop  that 
can  be  used  with  the  lens  so  as  to  produce  definition  up 
to  a selected  standard  of  excellence  all  over  a plate  of 
given  size,  the  objects  whose  images  are  seen  being  all 
equally  distant. 

“ A slow  lens  means  one  of  which  the  largest  normal 
stop  has  a less  diameter  than  has  C.I.  No.  6. 

“ A moderately  rapid  lens  is  one  of  which  the  largest 
normal  stop  is  C.I.  No.  6,  or  larger  than  that  size  and 
less  than  C.I.  No.  2. 

“ A rapid  lens  is  one  of  which  the  largest  normal  stop 
is  C.I.  No.  2,  or  larger  than  that  size  and  less  than  C.I. 
No.  2/3. 

An  extra  rapid  lens  is  one  of  which  the  largest 
normal  stop  is  C.I.  No.  2/3,  or  larger  than  that  size.” 

114.  Headings  of  Certificate,  1 — 8. — The  first  four 
headings  refer  only  to  the  numbering  of  the  lens,  the 
maker’s  name,  etc.,  and  need  not  be  further  considered. 

5.  Number  of  Reflecting  Surfaces. 

“ In  most  cases  the  number  of  reflecting  surfaces  of 
glass  is  known  at  once  from  the  type  of  lens,  but,  if  in 


228 


PHOTOGRAPHIC  OPTICS 


doubt,  a simple  experiment  will  settle  the  point ; the 
room  is  darkened,  and  the  reflection  of  a lamp  is 
observed  in  the  lenses ; each  of  the  surfaces  of  the 
lenses  will  give  one  direct  reflected  image,  and  the 
number  can  thus  easily  be  counted. 

6.  Centering  in  Mount. 

“ Two  different  errors  might  be  described  under  this 
heading  : either  (1)  the  optical  axis  of  a perfect  lens  may 
not  coincide  with  the  axis  of  the  mounting,  or  (2)  the 
axes  of  the  different  lenses  of  a doublet  or  triplet  may 
not  all  be  in  the  same  straight  line.  As  to  the  first  of 
these  errors,  we  believe  it  would  never  be  sufficient  to 
have  any  appreciable  effect  on  the  practical  value  of  a 
lens,  and  therefore  no  test  for  it  is  considered  necessary. 
With  regard  to  the  second  error,  Wollaston’s  test  is  the 
only  one  applied  ; this  consists  of  looking  at  the  flame 
of  a lamp  or  candle  through  a compound  lens,  and  noting 
if  all  the  different  images  of  the  light  as  seen  by  succes- 
sive reflections  from  the  surfaces  of  the  glass  can  be 
brought  into  line  by  a suitable  movement  of  the  whole 
lens,  which  should  be  the  case  if  the  component  lenses 
are  arranged  about  a common  axis. 

7.  Visible  Dejects^  such  as  Strice,  Veins^  Feathers,  etc. 

“ Under  this  heading  any  faults  detected  by  a careful 
inspection  are  given.” 

8.  Flare  Spot. 

The  nature  of  this  defect  has  been  explained  in 
§ 95  : -to  detect  it  the  lens  is  placed  in  an  ordinary 
camera,  which  is  pointed  to  the  sky  ; if  the  ground  glass 
is  brought  to  the  principal  focus,  the  flare  spot  is  then 
readily  visible. 

115.  Headings  of  Certificate,  9 — 16. — For  these  tests. 


LENS  TESTING 


2Q9 


Fig.  72. 


a testing  camera  or  apparatus  has  been  designed  by 
Major  Darwin  (see  Fig.  72). 


230 


PHOTOGRAPHIC  OPTICS 


“ The  three-legged  stool  or  bench,  seen  in  1,  represents 
the  legs  of  the  camera,  and  2 shows  the  apparatus  that 
takes  the  place  of  the  body  ; G is  the.  lens-holder,  and 
L M the  ground  glass,  both  of  which  are  capable  of 
independent  movement  backwards  and  forwards  on  the 
hollow  wooden  beam  D E,  called  the  ‘swinging  beam.’ 
There  is  a conical  brass  cap  or  pivot,  not  shown  in  the 
sketch,  under  the  upper  plank  of  the  swinging  beam, 
underneath  where  the  lens-holder  G is  shown  in  the 
sketch.  The  whole  of  the  apparatus  shown  in  2 is 
placed  on  the  top  of  the  three-legged  stool,  the  round- 
headed  iron  pin  A passing  loosely  through  a hole  in  the 
lower  plank  of  the  swinging  beam,  and  fitting  into  the 
conical  brass  cap  or  pivot.  The  swinging  beam,  being 
thus  supported  by  the  pin  A and  by  the  long  arm  B C 
of  the  stool,  is  capable  of  being  revolved  round  A as  a 
centre.  On  the  ground  glass  is  engraved  a horizontal 
line,  which  is  accurately  divided  into  fiftieths  of  an  inch  ; 
this  line  passes  through  the  centre  of  the  ground  glass 
(or  through  the  point  where  the  perpendicular  from  the 
lens-holder  cuts  the  glass),  and  is  also  parallel  to  B C, 
the  top  of  the  stool  on  which  the  swinging  beam  slides, 
when  the  camera  is  in  position ; thus  the  image  of  an 
object  will  appear  to  run  along  the  scale,  as  the  swing- 
ing bar  is  moved  from  side  to  side.  The  ground  glass 
can  be  brought  approximately  into  focus  by  means  of 
the  already -mentioned  movement  to  and  fro  on  the 
swinging  beam,  but  for  accurate  adjustment  a slow 
motion  arrangement  is  attached  to  the  movable  part 
itself ; the  handle  H gives  the  required  motion,  and 
there  is  a scale  S,  called  the  ‘ focus  scale,’  by  means  of 
which  these  small  movements  can  be  accurately 
measured.  On  the  lens-holder  there  is  a movement, 
corresponding  to  the  swing-back  of  an  ordinary  camera, 
by  which  the  lens  can  be  made  to  revolve  vertically 
round  a horizontal  axis,  without,  of  course,  any  corre- 
sponding movement  of  the  ground  glass  ; there  is  a 
vertical  arc,  Y,  by  means  of  which  we  can  read  off  the 


LENS  TESTING 


231 


vertical  angles  through  which  the  lens  is  rotated.  An 
arrangement  is  also  supplied  by  means  of  which  the  lens 
can  be  moved  backwards  and  forwards  on  the  movable 
stand,  thus  allowing  the  position  of  the  lens  to  be  so 
adjusted  that  the  horizontal  axis  can  be  made  to  pass 
through  any  point  in  its  axis. 


9.  Effective  Aperture  oj  Btops. 


Nmnber  I 

engraved  on  i 
stop. 

Effective 

aperture. 

Inches. 

//number. 

C.I.  Xo. 

No 

No 

No 

No 

No 

No 

No / 

“ The  effective  aperture  of  one  or  more  of  the  various 
stops  supplied  with  the  lens  is  found  by  a well-known 
method.  The  image  of  a very  distant  object  is  first 
brought  into  focus  on  the  ground  glass  of  the  testing 
camera  ; a collimator,  which  has  itself  been  previously 
focussed  on  a distant  object,  may  be  used  instead  of  the 
distant  object ; the  ground  glass  is  then  taken  out  and 
exactly  replaced  by  a tin  plate  with  a small  hole  at  the 
centre  ; this  hole,  which  should  be  very  small,  will, 
therefore,  be  at  the  principal  focus  of  the  lens.  The 
room  being  darkened,  a gas-burner  is  placed  behind  the 
small  hole,  and  thus  parallel  rays,  in  the  form  of  a 
cylinder,  are  made  to  issue  from  the  lens  towards  the 
front.  A piece  of  ground  glass,  with  a graduated  scale 
engraved  on  it,  is  now  held  in  front  of  the  lens,  and  the 
diaTiieter  of  the  illuminated  disc,  or  section  of  the 
cylinder  as  seen  on  the  glass,  is  directly  measured  off  as 
any  stop  is  inserted  in  its  place.  Thus  is  found  the 


232 


PHOTOGRAPHIC  OPTICS 


effective  aperture  of  the  largest  stop,  as  recorded  in  the 
Kew  Certificate  of  examination.  The  ratio  of  the 
effective  aperture  to  the  diameter  is  the  same  for  all 
stops  of  the  same  lens,  and  the  effective  aperture  of  the 
other  stops  is  either  measured  as  above,  or  calculated 
from  the  ratio  thus  found.  As  the  rays  are  parallel 
when  emerging  from  the  lens,  it  is  evident  that,  if  the 
stop  is  in  front  of  all  the  lenses,  the  effective  aperture 
will  be  the  same  as  the  diameter  of  the  stop  itself. 

“10.  Angle  of  Cone  of  Illumination  with  Largest  Stop  = 

giving  a Circular  Image  on  the  Plate  of 

inches  diameter.  Angle  of  Cone  outside  which 

the  Aperture  begins  to  he  eclipsed  with  Stop  C.I.  No. 

= giving  a Circle  on  the  Plate  of 

inches  diameter. 

Diagonal  of  Plate  = inches.,  requiring  a Field  of 

°(  = 2(|)). 

Stop  C.I.  No. is  the  Largest  Stop  the  whole  of  the 

Opening  in  which  can  he  seen  from  the  whole  of  the 
Plate:^’ 

The  meanings  of  the  terms  used  here  have  been 
explained  in  | 59  ; it  should  be  carefully  noticed  that 
the  angles  in  question  are  those  between  two  extreme 
pencils  on  opposite  sides  of  the  axis,  or  the  whole  vertical 
angles  of  the  cones. 

“The  outer  cone,  which  we  have  called  the  ‘cone  of 
illumination,’  gives  the  extreme  angle  of  the  field  of  the 
lens  without  regard  to  definition,  and  is  what  is  known 
to  French  authors  as  the  champ  de  visihilite.  To  find 
the  angle  of  the  cone  of  illumination,  the  lens  is  placed 
in  the  testing  camera,  and  the  observer  looks  through 
the  small  hole  in  a sheet  of  tin  plate  with  which  the 
ground  glass  has  been  replaced,  as  in  the  last  test ; the 
lens-holder  is  made  to  revolve  about  its  horizontal  axis, 
and  as  the  axis  of  the  lens  moves  away  from  zero,  first 
in  one  direction  and  then  in  the  other,  the  positions  at 


LENS  TESTING 


233 


which  all  light  appears  to  be  cut  off  are  noted  ; the  angle 
between  these  two  positions  as  read  on  the  vertical  arc, 
Y,  gives  the  angle  of  the  cone  of  illumination.” 

Before  making  the  test,  the  axis  of  rotation  should  be 
made  to  pass  through  the  nodal  point  of  emergence,  in 
the  manner  explained  in  § 105. 

The  angle  of  the  inner  cone,  that  is,  of  the  cone 
outside  which  the  opening  of  the  stop  is  partially 
eclipsed  by  the  mounting  of  the  lens,  etc.,  is  measured  in 
the  same  way  as  above  described  for  the  outer  cone,  and 
with  the  same  precautions.  When  looking  through  the 
small  hole,  the  positions  on  each  side  of  zero  at  which 
the  aperture  begins  to  be  shut  off,  and  beyond  which  it 
no  longer  appears  as  a perfect  ellipse,  are  easily  seen, 
and  the  angle  between  these  two  positions  as  measured 
on  the  vertical  arc  gives  the  angle  required.  The  angles 
of  these  two  cones  are  generall}^  given  when  the  observ- 
ation is  made  with  the  largest  stop  supplied  with  the 
lens. 

“ In  order  to  facilitate  the  consideration  of  the  cover- 
ing power  of  the  lens,  the  diameters  of  the  circles  which 
these  cones  make  by  cutting  the  photographic  plate, 
wlien  the  focus  is  adjusted  for  distant  objects,  are  given 
in  the  Certificate  of  Examination.  Having  found  the 
principal  focal  length  in  the  manner  to  be  described 
immediately,  the  size  of  these  circles  can  readily  be 
ascertained  by  a simple  graphical  method,  which  is 
hardly  worth  describing  in  detail. 

In  connection  with  this  test  it  may  be  convenient 
to  adopt  the  use  of  the  term  angle  of  field  under 
examination  (denoted  in  this  paper  by  2(^),  to  signify 
the  angle  subtended  at  the  nodal  point  of  emergence  by 
a diagonal  of  the  plate,  or  the  greatest  angular  distance 
which  could  be  included  in  the  photograph,  supposing 
the  focus  to  be  taken  on  a distant  object.” 

The  angle  is  found  either  graphically  or  by  the  method 
of  § 59  (end),  and  the  result  is  entered  on  the  certificate 
of  examination. 


234 


PHOTOGRAPHIC  OPTICS 


As  the  lens  is  to  be  examined  as  to  its  behaviour  with 
a plate  of  given  size,  the  converse  test  is  necessary ; the 
size  of  the  largest  stop  must  be  found  which  will  include 
the  given  plate  in  its  inner  cone. 

“ If  the  illumination  of  the  field  is  not  to  fall  off 
rapidly  towards  the  edges  of  the  plate,  for  the  normal 
use  of  the  lens  we  should  employ  a stop  which  covers 
(or  nearly  covers)  the  plate  of  the  given  size  with  its 
inner  cone ; that  is  to  say,  we  should  use  a stop  not 
larger  than  the  largest  stop  the  whole  of  the  opening 
in  which  can  be  seen  from  the  whole  of  the  plate.  In 
order  to  find  the  largest  stop  which  fulfils  the  above 
conditions,  the  lens  is  revolved  about  the  horizontal  axis 
until  the  vertical  arc  reads  half  the  angle  of  field  under 
examination,  and  then  the  different  stops  are  put  in  one 
by  one  until  the  largest  one  is  found  which  is  seen  not 
to  be  eclipsed  when  the  observation  is  made  through  the 
hole  in  the  tin  plate.  The  number  of  this  stop  is 
recorded  in  the  certificate. 


“11.  Principal  Jocal  length  = in. 

Back  focus,  or  length  from  the  principal  focus  to  the 
nearest  point  on  the  surface  of  the  lenses  = m.’’ 

The  principal  focal  length  is  found  with  the  testing 
camera  as  follows  : — The  nodal  point  of  emergence  being 
on  the  axis  of  rotation,  the  swinging  beam  is  brought 
approximately  to  a central  position.  The  two  iron  stops, 
T and  T',  are  fixed  so  as  to  allow  the  beam  to  swing  on 
either  side  of  zero,  through  an  angle  whose  tangent  is 
1/4  or  14°  2'.  A distant  object  is  focussed  on  the 
ground  glass,  and  the  testing  camera  is  arranged  so  that 
when  the  beam  is  approximately  in  a central  position, 
the  image  of  some  veil-defined  object  seen  through  a 
hole  in  the  window  shutters,  appears  on  the  central  line 
of  the  ground  glass. 

When  this  adjustment  is  made,  the  line  joining  F,  the 
centre  of  the  ground  glass,  to  the  centre  of  the  lens,  will, 


LENS  TESTING 


235 


if  produced,  pass  through  the  distant  mark.  When  the 
swinging  beam  is  moved  from  side  to  side,  the  image 
appears  to  run  along  the  ground  glass ; its  position  is 
noted  when  the  beam  is  in  contact  with  the  stops  T and 
T'.  We  shall  see  that  twice  the  distance  between  the 
points  so  noted  on  the  ground  glass  is  equal  to  the 
principal  focal  length  of  the  lens. 

Suppose,  to  begin  with,  that  the  lens  remains  still, 
but  that  the  position  of  the  object  is  varied  from  B to 


ITg.  73. 


C (Fig.  73),  so  that  the  image  on  the  ground  glass 
moves  from  to  C'.  If  No  are  the  nodal  points 
and  the  angle  C'  N^  F = 14°  2'.  “ 

Hence  C'  F - N^  F tan  C'  N^  F - N^  F tan  14°  2' 
= Ni  F/4 

. • . Ni  F = 4 C'  F = 2 C'  B' 
and  N|  F is  the  principal  focal  length. 

If  the  lens  is  revolved  and  the  object  is  kept  station- 
ary the  result  will  be  the  same,  for  the  motion  of  N2  is 
too  small  to  affect  the  size  of  the  angle  C'  N^  F. 

Hence  we  have  proved  that  the  principal  focal  length 


236 


PHOTOGRAPHIC  OPTICS 


is  equal  to  twice  the  length  which  the  image  travels 
along  the  scale  on  the  ground  glass. 

Major  Darwin  devotes  several  pages  to  the  discussion 


LENS  TESTING 


237 


of  the  accuracy  of  this  method,  and  concludes  that  if 
worked  with  reasonable  care  it  is  as  reliable  as  any  of 
the  other  methods,  and  it  has  the  advantage  of  being 
fairly  rapid. 


“ 12.  Curvature  of  the  Fields  or  of  the  Principal  Focal 
Surface.  After  focussing  the  p>late  at  its  centre^ 
movement  necessary  to  bring  it  into  focus  for  an 
image inches  from  its  centre  = inches. 

Ditto  for  an  object inches  from  its  centre  = 

inches. 

Ditto  for  an  object in.  from  its  centre  = in!^ 


Curvature  has  been  explained  in  § 75;  the  only 
question  to  be  considered  is  the  mode  in  which  it  is  to 
be  measured. 

Let  (Fig.  74)  A F B be  the  section  of  the  principal 
focal  surface  by  a plane  through  the  axis,  and  H F K 
the  section  of  a plate  touching  A F B at  F,  the  principal 
focus.  Then  the  picture  will  be  sharply  focussed  at  the 
centre,  but  at  points  like  f and  d it  will  be  out  of  focus; 
if,  for  instance,  we  wish  to  focus  sharply  at  c/,  we  must 
move  the  plate  forward  a distance  d c,  till  is  on  A F B. 

The  curvature  can  therefore  be  given  conveniently 
for  practical  purposes  by  stating  the  distances  like  c f 
f e through  which  the  plate  must  be  moved  in  order  to 
focus  sharply  at  points  such  as  d and  f. 

Another  method  of  measuring  the  curvature  of  a 
curve  like  A F B,  often  used  in  mathematical  work,  is  to 
give  the  radius  of  the  circle  which  will  most  nearly  fit 
the  curve  at  the  point  F ; the  curve  A F B is  not  as  a 
rule  exactly  circular,  but  a small  portion  near  F is  very 
approximately  so  and  we  can  calculate  its  radius.  It 
can  be  shown  that^  if  r is  the  radius 

( f F)^ 

T = — — L (very  approximately) 

2/e 

where  / is  a point  near  to  F. 

^ Williamson’s  Differential  Calculus-,  Ed.  4,  p.  291. 


238 


PHOTOGRAPHIC  OPTICS 


Example. — Take  the  case  given  in  the  certificate 
quoted  (heading  12). 

Here  y’F  = 1*5  inches,  y e = ‘02  inch. 

r = (l*5)^/’04  = 56*25  inches  = 4 ft.  8*25  in. 

Hence  the  section  of  the  portion  of  the  principal  focal 
surface  near  F is  very  approximately  a portion  of  a 
circle  of  radius  nearly  5 feet. 

[In  calculating  radius  of  curvature  the  measures  for 
a point  as  near  as  possible  to  the  axis  should  be  used.] 

“ The  following  is  the  method  of  finding  the  curvature 
of  the  principal  focal  surface.  The  image  of  a distant 
object  (or  of  the  collimating  telescope)  is  thrown  on 
that  point  on  the  ground  glass  where  the  axis  of  the 
lens  cuts  it,  the  focus  is  accurately  adjusted,  and  the 
focus  scale  is  read  off.  The  swinging  beam  is  then 
moved  so  that  the  image  comes  successively  to  positions 
at  convenient  intervals  from  the  centre  of  the  plate, 
and  on  each  occasion  the  focus  is.  adjusted  afresh,  and 
the  focus  scale  read  off.  By  subtracting  the  central 
reading  from  these  outer  readings,  the  results  recorded 
on  the  Certificate  of  Examination  are  obtained. 

“ 13.  Definition  at  the  Centre  with  the  Largest  Sto2)^ 

C.I.  Stoj)^  No. gives definition  over  the 

whole  of  a inch  by inch  plate. 

“ The  system  by  which  the  defining  power  is  measured 
consists  in  ascertaining  what  is  the  thinnest  black  line 
of  which  the  image  is  just  visible,  the  test  being  con- 
ducted in  the  following  manner.  The  test  object 
consists  of  a thin  straight  strip  of  steel,  about  0*1  inch 
wide,  and  about  an  inch  long ; it  is  capable  of  being 
rotated  about  an  axis  in  the  direction  of  its  greatest 
length,  thus,  if  seen  against  a bright  background,  making 
it  appear  as  a black  line  of  varying  width ; when 
presented  edgewise  to  the  objective,  it  is  so  thin  that 
the  image  becomes  invisible  ; and  there  is  an  arc  so 
graduated  that  the  angle  subtended  by  the  two  edges  of 


LENS  TESTING 


239 


the  strip  at  the  lens  can  be  at  once  read  off,  thus  giving 
a measure  of  the  apparent  thickness  of  the  line.  The 
test-object  is  placed  as  far  as  possible  from  the  lens  in  a 
darkened  room  (at  Kew  the  accommodation  in  this  respect 
leaves  much  to  be  desired),  and  beyond  it  is  a ground 
glass  screen  illuminated  by  a lamp. 

In  order  to  test  the  defining  power  of  a lens  in  the 
centre  of  its  field,  the  focus  is  first  very  carefully 
adjusted  on  the  ground  glass,  and  the  test-object  is  then 
slowly  revolved  from  the  edgewise  position,  where  its 
image  is  invisible,  until  the  first  appearance  of  a dark 
line  can  be  seen  against  the  bright  background  ; the 
angular  width  of  the  line  is  read  off,  and  is  noted  as  a 
measure  of  the  defining  power  of  the  lens  in  the  centre 
of  its  field.  The  light  of  the  lamp  is  regulated  so  that 
the  image  of  the  line  can  be  seen  as  soon  as  possible. 

“ Besides  measuring  the  defining  power  where  the 
axis  of  the  lens  cuts  the  focal  surface,  an  observation  is 
also  made  at  a point  representing  the  extreme  corner  of 
the  plate  of  the  size  for  which  the  lens  is  being  examined, 
that  is,  at  a distance  from  the  centre  equal  to  half  the 
diagonal  of  the  plate.  As  the  object  of  this  second 
test  is  to  measure  the  general  definition  over  the  whole 
plate,  the  focus  is  taken  at  a position  half-way  between 
the  point  of  observation  and  the  axis  of  the  lens,  this 
being  the  method  generally  adopted  by  practical  pho- 
tographers when  desirous  of  getting  the  best  general 
focus.  It  is  necessary,  moreover,  that  the  test-object 
should  be  so  arranged  that  the  steel  strip  makes  an 
angle  of  45°  with  the  horizon;  for,  since  the  diffusion 
of  the  image  near  the  margin  may  be  due  to  astigmatism, 
a false  impression  of  the  defining  power  will  be  obtained 
if  the  image  of  the  dark  line  coincides  in  direction  with 
either  of  the  focal  lines whereas  if  it  bisects  the  angle 
between  them,  as  will  then  be  the  case,  there  is  no  error 
in  the  result  from  this  cause.  The  test  is  not,  how- 
ever, conducted  in  quite  the  same  way  as  in  the  first 
instance ; the  test-object  is  set  at  a known  angle,  and 


240 


PHOTOGRAPHIC  OPTICS 


the  stops  are  slipped  in  one  after  another,  beginning 
with  the  largest  and  going  on  to  smaller  ones,  until  the 
image  of  the  black  line  on  the  bright  ground  is  first 
just  visible;  the  C.I.  No.  of  the  stop  with  which  the 
lens  gives  definition  up  to  a known  standard  at  the 
extreme  corner  of  the  plate  is  thus  ascertained,  and,  as 
it  may  fairly  be  assumed  that  the  definition  will  be  no 
worse  than  this  at  any  other  part  of  the  plate,  it  follows 
that  the  defining  power  over  the  whole  plate  comes  up 
to  or  exceeds  the  standard  selected. 

“ 14.  Distortion.  Deflection  or  sag  in  the  image  of  a 
straight  line  which^  if  there  were  no  distortion.,  would 
run  from  corner  to  corner  along  the  longest  side  oj 
a by plate  — 0* inch.^^ 

The  question  of  distortion  has  been  treated  at  length 
in  Chapter  III.;  it  should  be  noted  that  the  quantity 
given  in  the  Kew  certificate  is  not  the  same  as  that 
which  we  have  taken  as  the  measure  of  the  distortion. 
The  quantity  we  have  used  is  the  displacement  of  the 
point  in  question  from  the  position  it  should  occupy 
according  to  the  simple  theory,  and  it  is  this  that  is 
found  with  the  tourniquet.  But  in  the  Kew  test  the 
quantity  found  is  the  amount  by  which  the  image  of  a 
straight  line  sags  between  its  ends ; which  is  practically 
useful  in  giving  an  idea  of  the  curvature  of  the  image. 

“ The  following  is  the  method  adopted  at  Kew  of 
measuring  the  distortion  produced  in  the  image  by  the 
lens  under  examination.  Let  Fig.  75  be  a vertical 
section  through  the  testing  camera ; G G representing 
the  ground  glass ; F the  principal  focus ; and  N^  the 
horizontal  axis,  which  passes  through  the  nodal  point  of 
emergence,  the  adjustment  for  that  purpose  having 
already  been  made  for  test  No.  10.  The  lens-holder 
carrying  the  lens  is  first  turned  in  either  direction 
through  an  angle  /3,  such  that  C F,  or  I N^  tan  /3,  or 
/ tan  /3  is  equal  to  half  the  shortest  side  of  the  plate  for 


LENS  TESTING 


241 


which  the  lens  is  being  tested.  (The  horizontal  move- 
ment of  the  swinging  beam  in  the  testing  camera  gives 
an  easy  means  of  determining  the  angle  /3 ; a distant 
object  is  first  brought  to  focus  at  the  centre  of  the 
ground  glass,  and  then  the  swinging  beam  is  revolved 
about  the  axis  A (see  Fig.  72)  until  the  image  has 
moved  along  the  graduated  scale  a distance  equal  to 


Fig.  75. 


half  the  shortest  side  of  the  plate ; the  beam  is  thus 
made  to  move  through  the  angle  /3,  which  can  be  read 
off  with  sufficient  accuracy  on  B C,  the  top  of  the  wooden 
stool,  which  is  graduated  for  that  purpose.  After  this 
adjustment  has  been  made  the  ground  glass  is  brought 
into  focus  by  observing  the  image  of  a distant  object  at 
a point  P,  a little  below  C,  the  line  engraved  on  the 

R 


242 


PHOTOGRAPHIC  OPTICS 


glass  ; under  these  circumstances,  if  the  principal  focal 
surface  is  a plane,  and  if  the  lens  were  being  used  in 
the  ordinary  manner,  P P'  would  be  the  position  occu- 
pied by  the  photographic  plate,  the  section  shown  being 
taken  across  the  centre  of  the  plate  parallel  to  its 
shortest  side.  The  small  distance  P C is  carefully 
measured ; this  length  is  then  multiplied  by  secant  jS, 
thus  obtaining  C ' P,  which  we  will  call  a.  The  swing- 
ing beam  is  now  revolved  about  the  pivot  in  either 
direction,  so  that  the  image  moves  along  the  scale  on  the 
ground  glass  a distance  equal  to  half  the  longest  side  of 
the  plate  for  which  the  lens  is  being  examined  ; the 
sketch  in  Fig.  7 5 is  still  more  or  less  applicable,  C ' P ' 


Fig.  76. 


still  representing  a section  across  where  the  photographic 
plate  ought  to  be,  but  this  time  at  the  end  of  the  plate, 
not  at  its  centre  (F,  therefore,  no  longer  represents  the 
principal  focus)  ; in  fact,  what  has  been  done  is  to  make 
the  image  describe  what,  neglecting  distortion,  would  be 
a straight  line  from  the  centre  to  the  corner  along  the 
longest  edge  of  the  plate  : after  this  movement  has  been 
made,  the  length  of  C'  P is  again  obtained  by  measure- 
ment and  calculation,  and  this  time  let  the  result  be 
called  b ; the  operation  is  repeated  when  the  swinging- 
beam  is  revolved  to  an  equal  angle  on  the  other  side  of 
zero,  and  a third  length,  c,  is  thus  obtained.  In  Fig. 
76,  let  BAG  be  equal  in  length  to  the  longest  side  of 
the  plate,  and  let  a,  h,  and  c be  the  lengths  just  ob- 


LENS  TESTING 


24S 


tained  ; then  the  curve  hac  will  evidently  represent  the 
image  of  a straight  line  thrown  by  the  lens  under  exam- 
ination along  the  edge  of  the  longest  side  of  the  plate. 
Since  the  image  travels  along  a line  very  nearly  parallel 
to  the  engraved  line  on  the  ground  glass,  BAG  will  be 

nearly  parallel  to  the  chord  of  the  curve,  and  ^ ^ — a, 

2 

which  is  the  length  recorded  in  the  Kew  certificate,  will 
be  a very  close  approximation  to  the  sagitta  or  sag  of 
the  curve. 

“15.  Achromatism.  After  Focussing  in  the  Centre  of  the 
Field  in  White  Lights  the  Alovement  necessary  to 
bring  the  Plate  into  Focus  in  Blue  Light  {dominant 

wave-length  4420),  = 0* inch.  Ditto  in  Red 

Light  (dominant  wave-length  = 0’ inch.^''^ 

The  test  in  this  case  is  very  similar  to  that  described 
in  § 110,  but  it  is  made  by  eye  and  not  photographically. 

“ First  the  focus  is  carefully  adjusted  in  daylight  on 
a suitable  object  placed  as  far  away  as  possible  in  the 
room,  and  then  the  focus  scale  is  read  off.  After  this, 
a sheet  of  blue  glass,  the  colour  of  wliich  has  a dominant 
wave-length  of  4420,  is  placed  behind  the  object  and 
close  in  front  of  a small  opening  in  the  shutter  through 
which  all  the  light  enters  the  room ; the  focus  is  re- 
adjusted, the  focus  scale  read  off  again,  and  the  difference 
in  reading  to  that  observed  in  white  light  is  noted.” 

The  calculation  to  find  the  change  in  the  principal 
focal  length  is  very  similar  to  that  of  § 110;  here  it  is 
V that  is  varied  and  u that  remains  constant.  If  the 
symbols  have  the  same  meaning  as  before  it  can  be 
shown  in  a similar  manner  that — 


^ For  the  unit  in  which  w’ave-lengths  are  measured,  the  tenth 
metre,  see  § 6, 


244 


PHOTOGRAPHIC  OPTICS 


It  is  the  calculated  value  of  a which  is  entered  on 
the  certificate. 

A similar  process  is  then  performed  with  a sheet  of 
red  glass  the  colour  of  which  has  a dominant  wave- 
length of  6250. 

“ It  may  be  observed  that  either  the  principal 
focal  length  or  the  position  of  the  nodal  point  of 
emergence  may  vary  as  different  coloured  lights  pass 


Fig.  77. 

through  a lens.  It  would  not  be  difficult  to  investigate 
these  two  sources  of  error  separately,  but  the  results 
would  be  of  little  or  no  practical  value. 


“16.  Astigmatism,  Approximate  Diameter  of  the  Disc  of 
Diffusion  in  the  Image  of  a Pointy  with  stop  C.I. 

No. at inches  from  the  centre  of  plate  = 

0' inch, 

“The  following  is  the  method  of  examination  for 
astigmatism  : — The  room  is  darkened,  and  in  front  of 
the  lens  is  placed  a thermometer  bulb,  thus  obtaining, 
by  means  of  the  reflection  of  the  light  of  a small  lamp. 


LENS  TESTING 


245 


a fine  point  of  light.  The  lens-holder  of  the  testing 
camera  is  revolved  upwards  or  downwards  about  the 
horizontal  axis  so  that  the  axis  of  the  lens  makes  an 
angle,  0,  with  the  path  of  the  rays  coming  from  the 
thermometer  bulb ; the  angle  (j)  is  such  that  the  point 
of  observation  represents  the  extreme  corner  of  the 
plate  of  the  size  for  which  the  lens  is  being  examined ; 
that  is  to  say,  if,  in  Fig.  77,  G G represents  the  position 
of  the  ground  glass,  then  C P is  equal  to  half  the 
diagonal  of  the  plate  ; this  angle  has  already  been  found 
for  previous  tests.  If  the  lens  shows  any  astigmatism, 
the  image  of  the  point  of  light  can  be  made  to  appear, 
first  as  a fine  vertical  line,  and  then,  as  the  focus  is 
lengthened,  as  a fine  horizontal  line.  The  focal  scale 
is  read  off  at  each  of  these  positions,  and  the  difference, 
y,  between  the  two  readings  gives  a measure  of  the 
astigmatism.’’ 

Major  Darwin  then  shows  how  to  calculate  the  size 
of  the  patch  of  light  in  the  image  caused  by  this 
astigmatism. 


“17.  Illumination  of  the  Field.  The  figures  indicate  the 
relative  intensity  at  different  parts  of  the  plate. 

With  C.I.  Stop  No.  . With  C.I.  Stop  No.  . 

At  the  centre 100  : Ditto 100 

At  in.  from  the  centre  : Ditto 

At  in.  from  the  centre  : Ditto 

“ The  intensity  of  illumination  of  the  field  is  always 
greatest  near  the  axis  of  the  lens,  and  falls  off  more  or 
less  rapidly  towards  the  edges  of  the  plate.  The  lens 
should  therefore  be  examined  with  the  view  of  ascer- 
taining if  this  inequality  of  illumination  is  greater  than 
that  which  experience  shows  must  be  tolerated  under 
given  circumstances.  The  apparatus  employed  for  con- 
ducting this  test  is  shown  in  Fig.  78,  the  method  being 
devised  by  Captain  Abney.  There  is  a fixed  lamp  L, 


246 


PHOTOCxRAPHIC  OPTICS 


the  position  of  which  is  not  changed  during  the 
observations ; ¥ represents  a paper  screen,  placed  in 
that  position  in  order  to  give  a practically  uniform 
source  of  light ; O is  the  lens,  which  is  fixed  in  a frame, 
not  shown  in  the  sketch,  revolving  about  the  pivot  N ; 
by  means  of  a suitable  adjustment,  this  axis,  JST,  is 
made  to  pass  through  the  nodal  point  of  emergence  of 
the  lens.  At  S there  is  a sheet  of  cardboard  with  a 
small  hole  in  the  centre  at  H,  and  this  screen,  hole  and 
all,  is  covered  with  thin  white  paper  on  the  side  away 
from  the  lens  ; the  distance  between  H and  N is  always 


Fig.  78. 


made  equal  to  the  principal  focal  length  of  the  lens ; 
the  bar  D is  made  to  cast  a shadow  from  the  movable 
lamp  M on  the  paper  just  over  the  hole  in  the  card- 
board ; thus,  in  this  shadow,  the  paper  is  illuminated 
entirely  by  transmitted  light  from  the  lens,  whilst  the 
paper  round  it  is  illuminated  entirely  by  the  light  of 
the  movable  lamp. 

“ An  observation  is  made  in  the  following  manner  : — 
The  lens  is  first  placed  in  such  a position  that  its  axis 
passes  through  the  hole  H ; the  lamp  M is  then  moved 
backwards  or  forwards  until  the  transmitted  illumin- 


LENS  TESTING 


247 


ation  of  the  paper  at  H is  made  to  match  as  nearly  as 
possible  the  reflected  illumination  of  the  paper  round 
it  j the  distance  between  S and  M is  then  noted.  The 
lens  is  now  placed  in  the  position  s^own  in  Fig.  78, 
where  A B represents  the  length  of  the  diagonal  of  the 
plate  for  which  the  lens  is  being  examined,  and  where 
the  angle  (p  is  half  the  angle  of  field  under  examination. 
The  balance  of  light  is  readjusted  by  a movement  of 
the  lamp,  and  the  distance  M S is  read  off  a second  time. 
By  finding  the  inverse  ratio  of  the  squares  of  these 
two  readings,  we  obtain  the  ratio  between  the  illumin- 
ations at  P and  H,  the  lens  being  in  the  position  shown 
in  the  sketch,  and  the  object  being  supposed  to  be 
equally  illuminated  in  both  cases.  But  what  is  wanted 
is  the  ratio  between  the  illuminations  on  the  plate  at 
P and  A ; this  is  found  with  perfect  accuracy  by 
multiplying  the  ratio  of  the  illumination  at  P and  H, 
as  above  obtained  from  the  observations,  by  cos^  cp,  and 
this  result  is  that  which  is  entered  in  the  Certificate  of 
Examination.’’ 

The  reason  for  multiplying  by  cos^  cp  is  as  follows  : — 
The  difference  between  the  illuminations  at  H and  A 
is  due  to  two  causes,  first  the  different  distances  of  H 
and  A from  N,  and  secondly  the  obliquity  of  the  plate 
A B to  the  incident  light. 

Let  and  Ip  be  the  illuminations  at  H and  P,  and 
let  be  the  actual  illumination  of  the  plate  as  inclined, 
what  the  illumination  would  be  if  the  plate  were  at 
right  angles  to  the  incident  light;  then  (|  15,  a)  we 
have  seen  that 

Ia  = let  9 


Also  by  the  law  of  the  inverse  squares  (§  15) — 


= In  COS^  <P 


combining  the  two  results, 

cos  <p  = Ijj  cos^  cj) 


248 


PHOTOOr.APHIC  OPTICS 


which  is  the  required  result. 

The  extracts  from  Major  Darwin’s  paper  may  be 
terminated  by  the  concluding  paragraph  in  the  discus- 
sion of  the  last  test. 

“ In  connection  with  this  test  it  may  be  mentioned 
that  the  most  serious  omission  in  the  Kew  examination 
is,  that  there  is  nothing  to  show  the  actinic  transparency 
of  the  glass.  A slight  yellow  tinge  in  the  lenses,  which 
would  not  be  noticed  by  the  eye,  might  yet  be  sufficient 
to  seriously  affect  the  rapidity  of  the  objective.  But 


Fi(i.  79. 


no  test  could  be  devised  to  investigate  this  point  which 
did  not  introduce  photographic  methods,  and,  as  already 
stated,  the  consideration  of  expense  put  such  operations 
out  of  consideration  for  the  present.  I should  like,  if 
possible,  to  have  introduced  some  test  which  would 
have  at  the  same  time  indicated  the  actual  rapidity  of 
the  lens,  and  also  the  actual  falling  off  of  density 
towards  the  margin  of  the  photograph ; with  the  aid  of 
photography  this  would  not  have  been  difficult,  and  a 
plan  of  this  kind  would  have  been  adopted,  but  for  the 
cost.  This  subject  is,  liowever,  still  under  consideration 
by  Captain  Abney.” 

116.  Rapid  Test  of  a Lens. — In  the  Traite  Encyclo- 


LENS  TESTING 


249 


lyedique  de  Photographie^  is  given  a rapid  method,  due 
to  M.  Baille-Lemaire,  by  which  a photographic  test 
can  be  made ; the  information  furnished  is  not  so 
extensive  or  reliable  as  that  given  by  the  methods 
described  above,  but  this  method  has  the  advantage 
over  the  former  that  it  is  within  the  power  of  any 
photographer  to  execute  it.  The  arrangement  is  shown 
in  Fig.  7 9 ; a screen  ruled  with  horizontal  and  vertical 
lines  which  divide  it  into  squares  is  placed  so  that  the 


Eio.  80.  Fig.  81. 

axis  of  the  lens  passes  through  its  centre,  and  its  plane 
is  inclined  at  an  angle  of  45°  to  this  axis.  The  centre 
vertical  line  of  the  screen  is  then  focussed  sharply  and 
a photograph  taken  : the  appearance  of  the  screen  is 
shown  in  Fig.  80,  and  its  photograph  will  look  like  Fig. 
81  : the  letters  L,  P,  G,  D are  used  to  enable  the 
corresponding  portions  of  the  original  and  the  photograph 
to  be  easily  identified. 

1 Premier  Supidemcnt,  par  C.  Fabre,  p.  125,  Ed.  1892.  Gauthier- 
Villars,  Paris. 


250 


PHOTOGRAPHIC  OPTICS 


The  iwindijal  focal  length,  disregarding  the  nodal 
points,  can  be  found  roughly  by  measuring  the  corre- 
sponding distances  of  object  and  image  from  the  centre 
of  the  lens  and  calculating  in  the  usual  way.  The 
achromatism  is  tested  by  examining  if  the  vertical  line 
which  is  sharpest  in  the  photograph  is  the  same  as  that 
which  was  focussed  j if  it  is  not  so,  an  estimate  can  be 
made  of  the  distance  between  the  foci  for  visual  and 
actinic  rays,  by  noting  the  sharpest  line,  finding  the 
difference  between  the  distances  of  the  two  lines  from 
the  lens,  and  calculating  as  in  § 110. 

The  curvature  of  the  field  can  be  judged  by  joining 
the  sharpest  points  in  the  photograph  by  a continuous 
curve,  as  A O B in  Fig.  81  ; this  curve  is  roughly  a 
section  of  the  principal  focal  surface  by  a plane,  inclined 
at  an  angle  of  45°  to  the  axis  of  the  lens,  and  passing 
through  the  vertex  of  the  surface. 

The  depth  of  focus  can  be  examined  by  taking 
photographs  of  the  screen  while  the  ground  glass  is 
moved  slightly  and  the  screen  kept  still.  The  lens  can 
be  examined  for  distortion  by  placing  the  screen  at 
right  angles  to  the  axis  of  the  lens  and  examining  if 
the  lines  in  the  photograph  are  curved  or  not. 


CHAPTER  VI 


EXPOSURE,  STOPS,  AND  SHUTTERS 

117.  — In  this  chapter  will  be  considered  questions 
connected  with  exposure.  It  is  not  proposed  to  deal 
with  exposure  tables,  many  of  which  are  readily  acces- 
sible and  are  the  result  of  experiment,  but  rather  with 
the  relative  exposures  with  various  stops  and  lenses,  a 
knowledge  of  which  is  necessary  before  the  exposure 
tables  can  be  used  intelligently. 

118.  Relative  Exposures. — We  shall  assume  that 
plates  of  equal  sensitiveness  are  equally  exposed  (that 
is,  equal  chemical  effects  are  produced)  when  the  total 
quantity  of  light  received  by  equal  areas  are  equal,  no 
matter  how  long  a time  the  exposure  has  taken ; or  in 
other  words,  that  the  chemical  effects  depend  solely  on 
the  total  quantities  of  light,  and  not  on  the  manner  in 
which  they  have  been  received.  This  assumption  prob- 
ably is  not  strictly  true,  but  it  is  near  enough  to  the 
truth  for  practical  purposes. 

Hence,  when  using  plates  of  the  same  kind  with 
different  lenses  and  stops  we  have  to  find  the  times 
required  in  the  different  cases  for  the  plates  to  receive 
equal  quantities  of  light.  The  time  taken  for  the  plate 
to  receive  a given  quantity  of  light  is  inversely  propor- 
tional to  the  intensity  of  illumination  of  the  plate 

(§  13). 

Hence,  in  finding  the  ratio  between  the  times  of 
exposure  with  stops  or  diaphragms  of  various  apertures, 
we  shall  require  to  know  the  ratio  between  the  intensi- 
251 


252 


PHOTOGRAPHIC  OPTICS 


ties  of  illumination  of  the  plates  in  the  cases  to  be 
compared. 

119.  The  Illumination  of  the  Plate. — In  comparing 
intensities  of  illumination  we  shall  suppose  that  the 
object  in  each  case  is  of  the  same  brightness,  and  that 
it  consists  of  a plane  of  uniform  brightness  placed  at 
some  definite  distance  from  the  lens  and  at  right  angles 
to  the  axis ; this  will  simplify  our  ideas  and  will  not 
alter  the  result.  Let  us  consider  the  illumination  of  a 
small  circular  area  of  the  plate  having  its  centre  where 
the  axis  of  the  lens  cuts  the  plate  ; if  we  know  the 
illumination  in  this  region  and  the  relative  illumina- 
tions as  found  in  the  Kew  test,  No.  17  (§  115),  we  can 
find  the  illumination  at  any  other  part  of  the  plate. 

Let  (Fig.  82)  A B be  the  lens,  C D the  aperture  in 
the  stop  (the  lens  drawn  is  a simple  one  with  the  stop 
in  front,  but  the  theory  will  apply  to  lenses  of  all  kinds), 
X Y the  plane  object,  x y the  ground  glass  on  which  the 
image  is  focussed.  Suppose  the  lens  thin  and  its  centre 
o,  let  /be  its  focal  length,  u and  v the  distances  of  object 
and  image  respectively  from  o,  and  the  distance  of  the 
stop  from  o \ in  the  case  of  a compound  lens  u and  v 
will  be  measured  as  usual  from  the  nodal  points,  and  e 
will  be  the  distance  of  the  stop  from  the  nodal  point  of 
incidence.  Let  e/  be  the  radius  of  the  circular  disc 
whose  illumination  is  considered,  and  E F the  radius  of 
the  corresponding  disc  on  the  object ; we  shall  for  con- 
venience denote  the  discs  by  (E  F),  {e /).  Let  d be  the 
diameter  of  the  aperture  of  the  stop.  We  must  now 
inquire  how  the  illumination  of  {e f)  is  affected  by  the 
sizes  of  the  various  quantities  ; let  us  see  what  effect 
the  variation  of  the  various  quantities  will  have. 

All  the  light  that  reaches  (e f)  comes  from  (E  F),  but 
all  the  light  from  (E  F)  does  not  reach  {e /).  The 
quantity  of  light  from  (E  F)  that  reaches  (e f)  is  the 
whole  quantity  that  gets  through  the  aperture  C D ; 
this  depends  on  two  things,  the  illumination  at  C D pro- 
duced by  (E  F),  and  the  area  of  the  aperture  C D.  If 


EXPOSURE,  STOPS,  AND  SHUTTERS 


253 


the  distance  of  (E  E)  from  C D,  and  hence  the  illumin- 
ation, remains  constant,  then  the  illumination  of  {e f)  is 


254 


PHOTOGRAPHIC  OPTICS 


proportional  to  the  area  of  aperture  C D,  but  this  area 
is  77  cPj^  and  is  therefore  proportional  to  the  illumin- 
ation of  {e f)  therefore  varies  as  d^. 

Next  suppose  that  the  aperture  C D remains  un- 
changed, and  the  distances  of  object  and  image  are 
varied,  the  consequent  change  in  the  illumination  of 
{e f)  is  due  to  two  causes. 

(1)  The  distance  of  X Y from  C D varies,  and  con- 
sequently the  illumination  at  C D varies.  We  have 
seen  (§  14)  that  the  illumination  of  an  area,  due  to  a 
source  of  light,  varies  inversely  as  the  square  of  the  dis- 
tance ; hence,  on  this  account,  the  illumination  varies 
as 

(2)  The  area  of  the  object  to  which  the  area  {ef) 
owes  its  illumination  varies  as  the  distances  u and  v 
vary.  Let  A be  the  area  of  (E  E),  A'  that  of  {e f\  then 
by  similar  triangles  F E O,  /"e  0 — 


A 

A' 


f7  EF^ 
-n-ep 


= ^ or  A = -T.  A' 


Since  A remains  (by  hypothesis)  unchanged,  its  illumin- 
ation  will  be  proportional  to  A,  so  that  from  this  cause 
the  illumination  varies  as  A or  as 

Putting  together  these  two  effects  of  varying  u and 
Vy  we  see  that,  taking  both  of  them  into  account,  the 
illumination  of  {e  varies  as 

^2  1 

^ (u  — eY 

Now,  it  is  shown  in  works  on  Algebra  that  if  a 
quantity  A depends  on  two  others  a and  h,  and  if  it 
varies  as  a when  h is  constant,  and  as  h when  a is  con- 
stant, then  if  both  vary,  A varies  as  a and  h jointly  or 
A cc  ah.  So  h^re  if  I be  the  illumination  of  {ef)  and 
both  the  aperture  C D and  the  distances  u and  v are 
varied  together,  then 


IV 


1 


EXPOSURE,  STOPS,  AND  SHUTTERS 


255 


Now  the  time  of  exposure  varies  inversely  as  the 
ilumination,  for  the  greater  the  illumination  the  shorter 
is  the  time  that  is  required  for  a given  quantity  of 
light  to  fall  on  the  plate ; hence  if  T be  the  time  of 


exposure 


1 

T GC  -r  X X {u  — eY 


which  may  more  conveniently  be  expressed  by 

n,  ■ A _ 

T = 2 X “2  X 


where  X is  a number  whose  value  depends  on  the 
illumination  of  X Y and  the  sensitiveness  of  the  plate. 

This  expression  is  applicable  to  the  case  of  objects  at 
all  distances,  and  it  can,  if  required,  be  expressed 
entirely  in  terms  of  v,  the  distance  of  the  image  from 
the  lens ; in  many  cases  e may  be  neglected  in  compari- 
son with  u,  and  the  expression  then  simplifies  to 

M - A 


120.  Expression  when  the  Object  is  Distant. — 

When,  as  in  ordinary  outdoor  work,  the  object  is 
distant,  the  distance  v becomes  equal  to  the  principal 
focal  length  of  the  lens  (and  u being  great,  e can  be 
neglected),  which  gives  the  relation 


which  is  the  formula  usually  given ; expressed  in  words 
it  tells  us  that  the  time  of  exposure  is  proportional  to 
the  square  of  the  number  got  by  dividing  the  focal 
length  of  the  lens  by  the  diameter  of  the  aperture  of 
the  stop. 

121.  The  ftnantity  A . — As  stated  above,  the  quan- 
tity A depends  on  the  illumination  of  the  object  and  the 
sensitiveness  of  the  plate  ; the  latter  quantity  is  now 
usually  given  by  the  makers  of  the  plates  on  a scale 


256 


PHOTOGRAPHIC  OPTICS 


which  will  be  explained  later  (§  126),  but  the  former 
is  one  that  must  be  determined  by  the  judgment  of  the 
photographer.  Various  tables  have  been  published 
giving  methods  of  calculating  the  exposures  necessary 
for  various  kinds  of  objects  at  different  times  of  the 
year  and  at  different  times  of  day  and  with  different 
states  of  the  sky;  these  are  the  result  of  experiment, 
and  it  would  be  of  no  practical  use  to  theorize  about 
them.  Actinometers  have  been  invented  which,  by  the 
rate  of  blackening  of  a piece  of  sensitized  paper,  enable 
some  estimate  of  the  brightness  of  the  day  to  be  made, 
but  it  is  doubtful  if  they  are  of  much  practical  use,  for 
often  not  only  the  nature  but  the  situation  of  an  object 
has  to  be  considered,  and  the  proper  course  can  after  all 
be  determined  only  by  experience. 

122.  Sizes  of  Stops. — In  naming  the  sizes  of  the 
apertures  in  the  stops  of  a lens  it  is  convenient  to 
adopt  a system  in  which  the  name  will  give  a clue  to 
the  relative  exposure  required  with  the  stop.  Several 
systems  have  been  proposed,  })ut  the  one  which  has 
been  longest  established  and  seems  likely  to  die  hard  is 
as  follows  : — 

The  stops  are  denoted  by  the  ratio  f\d  of  the  focal 
length  of  the  lens  to  the  diameter  of  the  stop ; and  if, 
for  instance, //(i  =10,  then  the  stop  is  called  y/10. 

It  will  be  seen  at  once  from  § 120  that  the  times  of 
exposure  are  proportional  to  the  squares  of  these 
numbers;  thus,  for  instance,  the  exposure  with  the  stop 
yy20  will  be  four  times  as  long  as  with  stop y/lO,  and 
so  on. 

Example. — To  take  a certain  photograph  an  exposure 
of  ten  seconds  is  required  with  the  stop  // 1 2 ; find  the 
requisite  exposure  with  stop  y/32. 

Let  t be  the  time  required  in  seconds,  then 


10  “ \T2/  “ [jj  ~ 9’ 


640 


= 71  secs. 


The  disadvantage  of  this  system  is  that  to  find  the 


EXPOSURE,  STOPS,  AND  SHUTTERS 


257 


required  exposure  some  calculation  is  necessary,  which 
though  easy  enough  at  home  is  troublesome  in  the  field  ; 
it  is  more  troublesome  still  if  we  wish  to  find  the  size 
of  a stop  required  to  make  the  exposure  twice  or  three 
times  as  long,  for  then  we  should  have  to  extract  the 
square  roots  of  2 and  3. 

123.  Other  Systems  for  Stops. — Many  other  systems 
have  been  proposed  for  naming  stops,  the  principle  of 
them  being  that  the  number  denoting  the  stop  shall 
give  the  required  information  about  the  exposure  with 
very  little  calculation ; the  numbering  sometimes  in- 
creases and  sometimes  decreases  as  the  necessary  expo- 
sure increases. 

The  two  most  important  systems  are  those  of  the 
Photographic  Congress  at  Brussels  in  1889,  denoted  in 
the  Kew  Certificates  as  the  C.I.  system,  and  that 
adopted  by  the  Photographic  Society  of  Great  Britain 
and  denoted  U.S.ISr.  {Uniform  System  Numbers). 

In  the  C.I.  system  the  stop  y/10  is  taken  as  the 
starting-point  and  called  No.  1 ; the  time  of  exposure 
with  this  stop  is  taken  as  the  unit  exposure ; the 
remaining  stops  are  numbered  so  that  the  greater  the 
number  of  the  stop  the  longer  the  exposure  required ; 
No.  2 is  taken  to  require  double  the  exposure  of  No.  1 ; 
No.  3 three  times  that  of  No.  1,  and  so  on.  The  rule  to 
find  the  C.I.  number  of  a stop  is  to  divide  the  square  of 
the  focal  length  by  100  times  the  square  of  the  diameter 
of  effective  aperture  of  the  stop. 

In  the  U.S.N.  system  the  stop  y/4  is  taken  as  the 
starting-point,  and  then  the  stops  are  numbered  as  in 
the  C.I.  system,  so  that  the  time  of  exposure  is  propor- 
tional to  the  number  of  the  stop. 

Zeiss  has  his  own  system  of  marking  the  stops  of  his 
lenses  : he  takes  //1 00  as  the  starting-point,  and  numbers 
his  lenses  so  that  the  times  of  exposure  are  inversely 
proportional  to  the  numbers  of  the  stops. 

The  following  table  shows  the  connection  between  the 
three  systems ; the  numbers  given  are  enough  to  enable 

s 


S58 


PHOTOGRAPHIC  OPTICS 


an  idea  of  the  relative  values  to  be  formed  at  a 
glance. 

The  C.I.  numbers  are  not  worked  out  as  closely  as 
the  U.S.N.  system,  the  nearest  whole  number  in  the 
common  series  corresponding  to  whole  numbers  in  the 
C.I.  system  being  given. 


Table  showing  the  Connection  between  the 
Different  Systems  for  Naming  Stops. 


Ratio //(Z 
Comnion 
System 

C.I. 

System 

(Aiiprox- 

imate) 

U.S.N. 

Zeiss 

Ratio  f/d 
Common 
System 

C.I. 

System 

(Approx- 

imate) 

. U.S.N. 

Zeiss 

,//i 

1/10 

//25 

39-06 

16 

y/i-414 

1/8 

f/2S 

8 

49-00 

1/4 

//30 

56-25 

fl2-S28 

1/2 

//32 

10 

64-00 

>/3 

I/'562 

f/35 

12 

76-56 

1-00 

//36 

81-00 

8 

.//4'5 

1-26 

512 

//39 

15 

95-06 

1-56 

//40 

16 

100-00 

//o-656 

2-00 

//45 

20 

126-56 

>/6 

2*25 

//45-25 

128-00 

y/6-3 

2-47 

256 

//49 

24 

150*06 

y/7 

3-OG 

f/50 

156-25 

4 

y/y-l 

1/2 

3*07 

fl55 

30 

189-06 

V/8 

4-00 

y/56 

196-00 

3/4 

4*88 

32 

203-06 

.//8-8 

4*87 

fim 

225-00 

m 

5-06 

128 

fl%3 

40 

248-06 

y/lO 

I 

6*25 

y/64 

256-00 

//ll 

7*56 

//69 

48 

297-56 

//1 1-31 

8-00 

306-25 

'fll2 

9 00 

50 

315-06 

2 

>/12'5 

9-80 

64 

fp5 

56 

351-56 

//1-i 

1 ■ 2 

12 -25 

fpn 

60 

370-56 

fim 

lG-00 

//30 

64 

400-00 

fin 

3 

18-OG 

//88 

484-00 

>/18%> 

21-45 

32 

//!)() 

506-25 

4 

25 -OO 

//!)() -50 

512-00 

5 

30-25 

/'/»() 

576-00 

fl22-&2 

32-00 

.//lOO 

625-00 

1 

fl2i 

G 

30-00 

EXPOSURE,  STOPS,  AND  SHUTTERS 


259 


124.  Exposure  with  Dallmeyer’s  Telephotographic 
Lens. — The  principle  of  this  lens  is  that  a certain  por- 
tion of  the  image  formed  by  the  front  converging  lens 
is  picked  out  and  magnified  by  the  diverging  lens ; thus 
the  quantity  of  light  which,  in  a given  time,  falls  on 
any  particular  area  of  the  magnified  picture  is  the  same 
as  that  which  would  have  fallen,  in  the  same  time,  on 
the  corresponding  portion  of  the  picture  formed  by  the 
front  lens.  Thus  the  intensities  of  illumination  in  the 
two  cases  are  inversely  proportional  to  the  areas  of  the 
corresponding  pictures.  For  instance,  let  the  ratio  of 
the  linear  dimensions  of  the  two  pictures  (or  as  Dall- 
meyer  calls  it  the  magnification)  be  3i,  then  the  ratio 
of  the  two  areas  is  (32)*^  or  49/4  ; thus  the  light  which 
would  fall  on  a given  area  in  the  picture  formed  by  the 
front  lens  alone  has  to  cover  an  area  49/4  times  as  great 
in  the  magnified  picture. 

Hence  it  is  clear  that  the  exposure  for  the  magnified 
picture  will  be  49/4  times  as  long  as  for  the  picture 
that  would  be  formed  by  the  converging  combination 
alone. 

Reasoning  in  this  way  we  see  generally  that  the  time 
of  exposure  required  with  the  telephotographic  lens  can 
be  got  from  that  required  with  the  front  lens  alone  by 
multiplying  by  the  square  of  the  linear  magnification ; 
where  ‘‘linear  magnification”  has  the  meaning  given 
to  it  above. 

125.  Transparency. — In  the  preceding  sections  it  has 
been  assumed  that  the  glass  of  a lens  causes  no  loss  of 
light  either  by  absorption  or  reflection,  which  is  by  no 
means  the  case.  It  is  not  often  that  the  glass  is 
seriously  coloured,  but  in  every  lens  light  is  lost  by 
reflection  and  scattering  at  every  surface  (|  10),  and 
objectives  with  few  surfaces  let  through  more  light  than 
do  complicated  objectives. 

Hence  in  the  expression  for  the  time  of  exposure  in 
I 119,  the  quantity  A will  depend  on  the  nature  and 
construction  of  the  lens  as  well  as  on  the  other  quantities 


260 


PHOTOGRAPHIC  OPTICS 


enumerated  in  | 121.  The  expression  T = Xv’^/d^  can- 
not therefore  be  relied  on  to  compare  the  times  of 
exposure  with  different  lenses  ; but  it  will  in  most 
cases  give  a very  fair  idea  of  a required  exposure.  If 
two  or  three  lenses  are  being  constantly  used,  the 
operator  will  in  a short  time  be  able  to  form  a fair  idea 
of  the  relative  speeds,  and  to  find  out  how  to  modify 
the  exposures  found  from  the  formula. 

It  is  not  hard  to  make  a photographic  examination 
of  the  relative  powers  of  lenses  by  photographing  the 
same  object  with  constant  illumination,  trying  various 
exposures  with  the  different  lenses ; the  experiment  can 
be  made  with  an  ordinary  camera  and  slide.  The 
object  should  be  a uniformly  illuminated  object  of  some 
size,  such  as  a white  wall  on  a fairly  bright  day ; focus 
the  object,  and  find  as  closely  as  possible  the  proper 
exposure  with  one  of  the  lenses  to  be  examined.  In 
the  final  test  the  same  plate  must  be  used  with  the 
different  lenses,  to  avoid  differences  in  the  development ; 
this  can  be  done  without  trouble. 

When  using  the  lens  for  which  the  proper  exposure 
is  known,  draw  the  slide  to  expose  only  one-third  of 
the  plate,  and  use  a small  stop  to  make  the  exposure 
required  as  long  as  possible.  Now  take  the  slide  to 
the  dark  room  and  turn  the  plate  round,  end  for  end, 
so  that  if  the  slide  is  now  drawn  it  is  the  unexposed 
portion  which  is  first  exposed.  Then  fix  the  second 
lens  to  the  camera  and  use  the  stop  corresponding  to 
that  used  in  the  first  case,  so  that  if  the  lenses  were 
similar  the  exposures  required  should  be  equal ; suppose 
the  second  lens  is  slower  than  the  first.  Draw  the 
slide  to  uncover  nearly  all  the  unused  portion  of  the 
plate,  and  expose  for  the  same  time  as  with  the  first 
lens  ; then  push  in  the  slide  to  hide  a strip  of  the 
exposed  plate,  and  expose  for  a short  time  longer,  say 
half  a second  ; again  push  in  the  slide  to  hide  another 
strip  of  the  plate,  and  expose  for  another  half-second  ; 
and  so  on  till  three  or  four  exposures  have  been  made. 


EXPOSURE,  STOPS,  AND  SHUTTERS 


261 


The  effect  is  that  different  strips  of  the  plate  are 
exposed  for  different  times,  and  can  easily  be  recognized 
on  development.  Develop  the  plate  and  then  compare 
the  densities  of  the  strips  taken  with  the  second  lens 
with  that  of  the  part  taken  with  the  first  lens.  The 
strips  of  density  equal  to  that  of  the  first  part  will  show 
what  exposure  with  the  second  lens  is  equal  to  that 
with  the  first.  From  the  result  obtained  the  correction 
to  be  made  in  calculations  of  relative  exposures  can  be 
calculated. 

If  a very  exact  determination  is  required,  the  relative 
illumination  at  different  parts  of  the  plate,  as  found  by 
the  Kew  test,  must  be  taken  into  consideration.  The 
exposures  should  be  made  as  long  as  possible  to  make 
the  method  capable  of  giving  good  results,  for  it  is 
impossible  to  give  a shorter  exposure  by  hand  than  half 
a second,  with  any  certainty ; and  if  the  exposure  with 
the  first  lens  was  only  one  or  two  seconds,  an  error  of 
even  one-fourth  second  ma}^  make  a large  difference  in 
the  result. 

Example. — The  speeds  of  two  lenses  are  to  be  com- 
pared, the  stop  y/40  is  used  with  each  ; the  exposure 
with  the  first  lens  is  12  seconds,  and  that  with  the 
second  lens  is  found  by  experiment  to  be  lOi  seconds. 

Here,  since  corresponding  stops  were  used,  the  ex- 
posures required  in  the  two  cases  should  have  been 
equal,  had  the  lenses  been  quite  similar ; the  longer 
exposure  required  with  the  second  lens  is  due  to  the 
larger  loss  of  light.  Equal  effects  are  produced  by 
equal  quantities  of  light,  so  that  the  ratio  of  the 
illumination  produced  by  the  second  lens  to  that  pro- 
duced by  the  first  is  12/lOj  or  24/21  or  *87. 

Hence  the  exposures  required  with  the  first  lens  are 
onl}^  *87  of  the  corresponding  exposures  with  the  second 
lens. 

126.  Sensitometers  and  Sensitometer  Numbers. — To 

compare  the  sensitiveness  of  different  plates,  we  require 
evidently  a constant  source  of  illumination,  and  a means 


262 


PHOTOGRAPHIC  OPTICS 


of  exposing  the  plate  to  certain  definite  portions  of  this 
illumination.  The  sensitometer  most  generally  known 
and  used  is  that  of  Warneke.  The  scale  of  the  instru- 
ment consists  of  a plate  of  glass  composed  of  twenty-five 
different  pieces,  tinted  so  as  to  be  of  constantly  increas- 
ing opacity ; on  each  piece  of  glass  is  placed  an  opaque 
number.  This  scale  is  placed  in  a special  holder,  in 
contact  with  the  plate  to  be  tested,  a sheet  of  black 
paper  being  placed  behind  the  plate. 

The  source  of  light  employed  is  a phosphorescent  plate 
of  sulphate  of  calcium  ; to  excite  the  phosphorescence  a 
magnesium  ribbon  about  3 cm.  long,  *015  cm.  thick, 
and  *2  cm.  breadth  is  burned  as  near  as  possible  to 
the  plate.  Immediately  the  magnesium  is  burned,  60 
seconds  are  counted,  and  during  the  30  seconds  which 
follow,  the  luminous  plate  is  placed  on  the  scale ; the 
plate  is  then  developed. 

When  developed  the  plate  shows  the  numbers  on  the 
squares,  the  tints  of  which  get  fainter  and  fainter ; the 
last  number  visible  represents  the  sensitiveness,  and  is 
given  as  the  sensitometer  number. 

In  comparing  two  plates  care  must  be  taken  to 
develop  them  under  the  same  conditions  and  with 
identical  proportions  of  fresh  solution.  Since  the 
Warneke  sensitometer  was  devised  the  sensitiveness  of 
plates  has  been  increased,  and  the  original  scale  is  not 
long  enough  ; either  the  scale  is  extended,  or  makers 
make  an  estimate  from  what  is  shown  by  the  old  scale. 
In  Fig.  83  is  shown  the  result  of  the  test  on  a Sandell 
II.  plate;  the  sensitiveness  is  estimated  at  28. 

Hurter  and  Driffield^  have  published  a careful  in- 
vestigation on  the  action  of  light  on  sensitive  plates, 
and  have  devised  a method  for  the  estimation  of 
sensitiveness ; their  paper  is  well  worthy  of  careful 
study.  It  is  impossible  to  give  here  more  than  a short 
abstract  of  the  part  which  concerns  the  subject  in  hand. 

^ Journal  of  the  Society  of  Chemical  Industry.  No.  5.  Yol.  ix. 
May  31,  1890. 


EXPOSURE,  STOPS,  AND  SHUTTERS 


263 


They  take  as  the  density  a quantity  which  is  pro- 
portional to  the  amount  of  silver  reduced  by  the  action 
of  the  light  and  of  the  developer,  per  unit  area  of  the 
sensitive  film ; this  differs  from  Abney’s  definition 
(§"  19).  They  have  devised  a special  form  of  photo- 
meter, with  the  scales  arranged  to  read  off  the  density 
directly.  A series  of  experiments  was  made  in  which 
portions  of  the  same  plate  wore  exposed  for  various 
times  and  then  developed;  the  densities  were  then 


Fig.  83. 


measured.  It  was  found  that  the  whole  time  of  ex- 
posure might  be  divided  into  four  periods  : first  came 
the  period  of  under-exposure,  during  which  the  density 
increased  very  slowly  with  the  exposure,  but  was 
nearly  proportional  to  it ; during  the  second  period  the 
density  increased  much  more  rapidly  with  the  exposure, 
and  the  contrasts  produced  were  in  consequence  sharp, 
this  was  called  the  iieriod  of  correct  representation ; 
in  the  third  period  the  density  again  increases  slowly 


264 


PHOTOGRAPHIC  OPTICS 


with  the  exposure  up  to  a maximum,  this  is  the  period 
of  over-exposure  ; finally  the  last  period  is  that  during 
which  the  density  diminishes  with  the  exposure,  and 
reversal  takes  place. 

It  was  found  both  from  theory  and  experiment  that 
if  the  exposure  lies  in  the  second  period,  the  connection 
between  the  density  D,  obtained  with  a given  time  of 
exposure  of  t seconds,  is  of  the  form 

D = ^ log  • • • (^0 

where  I is  the  intensity  of  the  light  incident  on  the 
plate,  and  k and  i are  constants  depending  on  the 
nature  of  the  plate. 

If  two  experiments  are  made  we  can  from  these 
calculate  the  values  of  k and  i,  and  then  use  the  formula 
to  determine  the  densities  produced  by  other  exposures. 

The  intensity  I of  the  light  incident  on  the  plate  is 
measured  in  terms  of  that  produced  by  a standard 
candle,  placed  at  a distance  of  1 metre  from  the  plate ; 
the  exposure  E,  or  the  total  quantity  of  light  which 
falls  on  the  plate,  is  proportional  to  the  product  of  I 
into  the  time  of  exposure,  so  that  we  may  put  E = I j^, 
and  write  the  above  relation 

B = . . (b) 

If  the  time  is  measured  in  seconds,  the  unit  of 
exposure  here  used  is  called  a candle-metre  second,  or 

c.m.s. 

The  quantity  i was  called  the  inertia  of  the  plate. 
If  two  plates  of  inertias  i and  are  to  be  impressed 
with  the  same  density  by  exposure  to  light,  whose 
intensity  is  the  same  in  each  case,  for  times  t and 
then  will 

It  _ I t _ ty 

i i 

for  then  only  would  D have  the  same  value  for  each. 


EXPOSURE,  STOPS,  AXD  SHUTTERS 


265 


This  means  that  to  produce  similar  effects  on  two  plates, 
the  times  of  exposure  must  be  proportional  to  their 
inertias,  and  hence  a knowledge  of  the  inertia  of  a 
plate  will  enable  us  to  form  an  estimate  of  its  rapidity. 

Tn  order  to  find  the  quantity  i for  a plate  : “We 
give  to  the  plate  at  least  two  exposures  falling  within 
the  period  of  correct  representation  and  develop.  We 
then  measure  the  densities  exclusive  of  fog.  We  thus 
obtain  two  equations  connecting  the  two  densities 
and  D2  with  the  two  known  exposures  and  E2,  viz.  : 


Di  = A;  log  ^ and  k log  ?? 

i i 

from  which  we  obtain  by  elimination 

_ . D.7  log  E,  — Di  log  Eo  ” 

~ ' D,-D. ‘ 

Hence  the  value  of  log  i can  be  calculated  and  the 
value  of  i found. 

In  * practice  the  central  portion  only  of  the  plate 
should  be  used,  as  the  film  is  liable  to  be  of  unequal 
thickness  at  the  margin.  In  order  to  ensure  at  least 
two  exposures  falling  within  the  period  of  correct 
representation,  eight  exposures  of  2*5,  5,  10,  20,  40,  80, 
160,  and  320  c.m.s.  (candle-metre  second  units)  are 
given  ; a strip  of  plate  is  left  unexposed,  but  is  developed 
in  order  to  make  allowance  for  any  fogging  that  may 
occur.  Too  great  density  is  avoided,  but  a decided 
deposit  is  obtained  for  the  lower  exposures. 

Exam'ple, — With  a certain  plate  the  following  measures 
were  made  : — 


Exposures  .... 

2-5 

5 

10 

20 

40 

80 

1-01 

55  -2 

160 

Densities  .... 
Differences .... 

•085 

•0 

•175 
9 -07 

•250 

'5  -2] 

•460 

LO  -21 

•755 

15  *2 

1-27 

:60 

On  looking  at  the  differences  between  the  densities 
for  the  various  exposures  we  see  that  the  exposures 


266 


PHOTOGRAPHIC  OPTICS 


2 '5,  5,  and  10  c.m.s.  lie  in  the  first  period,  and  that 
exposures  20  to  160  c.m.s.  lie  within  the  period  of 
correct  representation.  Choosing  exposures  20  and  160 
for  calculation  we  get 


log  i 


1-27  X log  20  - -460  X log  160 
1-27  - *460 


•787 


Hence  i = 6 '12. 

The  speed  of  the  plate  is  the  inverse  of  the  quantity 

1,  for  the  greater  the  speed  the  shorter  should  be  the 
exposure. 

The  quantity  which  is  quoted  by  Hurter  and  Driffield 
as  the  speed  is  the  value  of  34/^  ; for  instance,  for 
three  grades  of  plates  of  a certain  make  called  ordinary, 
rapid,  and  extra  rapid,  the  values  of  i were  found  to  be 

2,  1*4,  and  *56;  the  speeds  are  17,  24,  and  60  respec- 
tively. 

The  calculation  of  relative  exposures  with  these 
numbers  is  made  in  the  same  way  as  with  those  of  the 
Warneke  system. 

Example, — With  the  ordinary  plate  just  mentioned  the 
time  of  exposure  required  was  5 seconds ; find  the  time 
of  exposure  required  with  the  extra  rapid  plate  ; let  t 
be  the  time  of  exposure  required,  then — 

t n , 

- = — ov  t = 1*42  sec. 

5 60 


Hurter  and  Driffield  at  the  beginning  of  their  paper 
assume  that  with  a given  thickness  of  film,  the  proportion 
of  incident  light  that  is  stopped  is  proportional  to  the 
quantity  of  silver  precipitated  per  unit  area.  For 
instance,  that  if  a certain  quantity  of  silver  cuts  off  one 
quarter  of  the  incident  light,  then  double  the  quantity 
of  silver  will  cut  off  one  half  of  the  light,  the  thickness 
remaining  constant.  The  correctness  of  this  assumption 
is  open  to  doubt ; in  cases  where  the  quantity  of  silver 
is  small,  so  that  the  particles  are  not  crowded,  it  is  most 
likely  true,  but  if  there  is  a large  quantity  of  silver 


EXPOSURE,  STOPS,  AND  SHUTTERS 


267 


already  present,  so  that  nearly  all  the  light  is  intercepted, 
then  if  more  silver  be  introduced  it  is  very  likely  that 
some  of  it  will  merely  lie  behind  that  already  present 
and  not  add  to  the  opacity. 

On  the  other  hand,  the  silver  is  reduced  in  the  film 
by  the  action  of  light,  and  hence  all  the  silver  present 
will  probably  be  in  such  a position  as  to  intercept  light. 
So  that  though  the  assumption  may  not  be  true  in 
general,  yet  in  the  special  case  of  photography  it  is 
probably  trustworthy. 

127.  Exposure  for  Objects  in  Motion. — When  photo- 
graphing objects  in  motion,  there  are  considerations 
other  than  those  of  the  sensitiveness  of  the  plate  that 
have  to  be  attended  to  in  estimating  the  exposure ; for 
if  the  exposure  be  too  long  the  object  will  have  moved 
over  a sensible  distance  on  the  plate  and  blurring  will 
result. 

All  rapid  exposures,  though  commonly  called  in- 
stantaneous, last  for  a definite  time,  and  during  that 
time  the  image  of  the  moving  object  is  moving  across 
the  plate  ; if  the  line  traced  by  any  point  of  the  image 
exceeds  a certain  length  it  can  be  seen  as  a line,  but  if 
less  than  that  length  it  will  be  to  the  eye  indistinguish- 
able as  a point. 

It  is  impossible  to  state  this  length  exactly,  but  it 
may  roughly  be  taken  to  be  1/100  inch  ; if  the  distance 
from  which  the  picture  is  to  be  viewed  is  large,  it  may 
be  made  greater  with  safety. 

To  enable  an  estimate  of  the  largest  possible  exposure 
to  be  made,  we  must  find  the  speed  at  which  the  image 
traverses  the  plate,  when  we  know  the  velocity  of  the 
object. 

First  let  the  object  be  moving  parallel  to  the  plate 
(Fig.  84),  let  A B be  the  distance  moved  by  a point  of 
the  object  in  one  second,  and  a h the  corresponding- 
distance  moved  by  the  image  on  the  plate.  Let  n and 
If  be  the  distances  of  object  and  image  from  the  lens,  and 
let  Y be  the  velocity  of  the  object  in  inches  per  second. 


268 


PHOTOGRAPHIC  OPTICS 


then  A B = Y inches ; let  f be  the  principal  focal 
length  of  the  lens. 


EXPOSURE,  STOPS,  AND  SHUTTERS 


269 


Then  by  similar  triangles 
ah 

Xb 


U 7 ^ A -r»  ^ TT  • 1 

___  = ab=  - AH  = - . V inches. 

O IS  u u 


1 1 1.1  11  « +./ 
but = 7>  • • “ = - + 7 = T 

V U J V U J %IJ 


. * . ah  = 


/ 


^ +f 


. V inches 


{a) 


Example. — The  object  is  a train  moving  at  the  rate 
of  30  miles  an  hour,  and  it  is  distant  60  feet ; the 
focal  length  of  the  lens  is  6 inches. 

Here  30  miles  an  hour  = 44  feet  per  second. 


. *.  V = 44  X 12  inches  per  second,  u — 60,  / = — 6 


1/^ 

60  - 1/2 


X 44  X 12  = 


44  X 12 
119 

— 4*4  inches. 


Hence,  in  this  case  the  image  would  move  4*4  inches 
on  the  plate  in  one  second,  and  to  get  a sharp  picture 
the  exposure  must  not  be  much  more  than  one  five- 
hundredth  of  a second. 

The  negative  sign  means  that  object  and  image  move 
in  opposite  directions. 

The  accompanying  table  (p.  270),  calculated  by  Mr. 
Henry  Tolman,  shows  the  number  of  inches  through 
which  the  image  moves  on  the  ground  glass,  in  one 
second,  when  the  object  is  moving  with  various  velocities 
and  is  30,  60,  or  120  feet  distant  from  the  camera; 
the  focal  length  of  the  lens  being  6 inches. 

The  distances  moved  through  with  lenses  of  dif- 
ferent focal  lengths  may  be  found  approximately  from 
this  table,  for  the  distance  is  nearly  proportional  to  the 
focal  length,  as  an  examination  of  expression  (a)  above 
will  show. 

If  the  motion  is  inclined  at  an  angle  0 to  A B (Fig. 
54)  the  result  will  not  be  quite  the  same  as  in  the 
former  case.  Let  the  object  move  along  inclined  at 


270 


PHOTOGRAPHIC  OPTICS 


B to  A B,  with  velocity  V inches  per  second,  the 
resolved  part  of  its  velocity  parallel  to  A B is  shown  to 
be  Y C06'  0]  we  must  use  this  in  place  of  Y and  the 
relation  becomes 


ah  = 


/ 

^ +y 


. Y cos  0 


Lens  6 in.  Equiv.  Focus,  Ground  Glass  at  Principal  Focus 
OF  Lens. 


Miles  i»er 
Hour. 

Feet  l er 
Second. 

Distance  on  Ground 
Glass  in  inches  witli 
Object  30  ft.  away,  in 
one  second. 

Same  with 
Object  60  ft.  ! 
away. 

Same  with 
Object  120  ft.  ' 
away. 

1 

I 

U 

•29 

•15 

•073 

2 

3“ 

•59 

•29 

•147 

3 

44 

•88 

•44 

•220 

4 

6^ 

I-I7 

•59 

•293 

5 

74 

1-47 

•73 

•367 

6 

9 

1-76 

•88 

•440 

7 

104 

2-05 

1-03 

•513  1 

8 

12 

2-35 

1-17 

•587  1 

9 

13 

2-64 

1-32 

•660 

10 

144 

2-93 

1-47 

•733 

II 

le'^ 

3-23 

1-61 

1 -807 

12 

174 

3-52 

1-76 

•880 

13 

19 

3-81 

1-91 

•953 

14 

204 

4 -I  I 

2-05 

1-027 

15 

22 

4-40 

2-20 

1-100 

20 

29 

5-87 

2-93 

1 -467  I 

25 

37 

7-33 

3-67 

1 -833  i 

30 

44 

8-80 

4-40 

2-200  1 

35 

51 

10-27 

5-13 

2-567 

40 

59 

11-73 

5-97 

2 933 

45 

66 

13-30 

6-60 

3-300  . 

50 

73 

14-67 

7*33 

3-667 

55 

80 

16-13 

8-06 

4-033 

60 

88 

17-60 

8-80 

4-400 

75 

no 

22-00 

11-00 

5-500 

100 

147 

29-33 

14-67 

7-333 

125 

183 

36-67 

' 18-33 

9-167 

150 

220 

44-40 

1 22-00  . 

11-000 

EXPOSURE,  STOPS,  AND  SHUTTERS 


271 


Example. — Take  the  same  data  as  in  the  last  example, 
except  that  the  motion  is  inclined  at  an  angle  of  60°  to 
the  former  direction,  then 

ah  = — 4*4  6*06'  60°  = ■—  4*4  x ‘5  = — 2*2  in. 
or  the  motion  now  is  only  half  of  what  it  was  before. 

128.  Object  moving  Towards  or  Away  from  the 
Camera. — Such  a case  as  this  occurs  when  an  express 
train  is  photographed  from  a bridge  under  which 
it  passes  ; the  image  then  remains  fairly  still  on  the 
plate,  but  grows  larger  or  smaller  as  the  object 
advances  or  recedes. 

Using  the  same  figure  as  before  (Fig.  84)  let  A B 
be  the  object  receding  from  the  lens ; let  A N = x 
inches,  and  let  A B recede  inches  in  one  second.  We 
have  seen  that 

h n = — • AN  = — — ^,x  inches. 

^ + y f 

Let  hpi  be  the  length  of  h n after  one  second,  then, 
since  the  object  is  now  distant  %ib  -{•  % inches 

h.  n = — .X’ inches. 

u J 

The  difference  between  u and  h u is 
f X J X _ f z 

n+f  M + a + /’  {u  +/)  {u  +7  + «)  ^ 

or  if  can  be  neglected  on  comparison  with  u f in 
the  denominator,  then 

. / 

(u  + 

Example.— K railway  engine,  breadth  5 feet,  moving 
at  30  miles  an  hour,  is  at  a distance  of  200  feet  from 
the  camera ; the  focal  length  of  the  lens  is  6 inches. 

Here  we  may  neglect  compared  with  u and  get 

Distance  moved  = « x 


Distance  moved  by  h 


/) 


o-  * 


• (&) 


272 


PHOTOGRAPHIC  OPTICS 


Now  f = — z — 44x12  inches,  a;  = 30  inches, 
u = 200  X 12  inches. 


Distance  moved  = 
•—  *0165  inch. 


6 X 44  X 12 
(200  X 12)2 


X 30  = 


Hence  the  edge  of  the  image  of  the  train  moves  at 
the  rate  of  *0165  inch  per  second,  and  the  negative 
sign  means  that  the  size  of  the  image  is  decreasing  as 
it  should  do. 

On  comparing  the  expressions  we  notice  that  in  (6) 
there  is  the  square  of  {u  + f)  in  the  denominator, 
while  in  (a)  the  first  power  occurs;  since  (^  +/*)  is 
fairly  large  compared  with  the  other  quantities  involved, 
we  see  that  the  motion  on  the  plate  will  be  much  less 
when  the  object  advances  directly  towards  the  camera 
than  when  it  is  moving  parallel  to  the  plate. 


Shutters. 

129.  — When  a short  exposure  has  to  be  made  some 
mechanical  device  is  required  to  uncover  and  cover  the 
lens,  the  hand  cannot  make  an  exposure  much  under 
a quarter  of  a second.  Many  forms  of  shutters  have 
been  designed  and  are  well  known  to  practical  photo- 
graphers ; we  do  not  propose  to  give  descriptions  of  the 
various  forms,  but  to  enumerate  two  or  three  classes  in 
which  many  shutters  can  be  placed,  and  to  examine  the 
principles  which  apply  to  their  use. 

130.  — Duration  of  Exposure. — Many  shutters  which 
are  worked  by  springs  are  marked  by  the  makers  to 
indicate  the  times  of  exposure  with  given  adjustments ; 
as  the  numbers  given  are  not  always  reliable  it  is  well 
to  be  able  to  test  them,  which  can  be  done  without 
much  trouble. 

(1)  If  the  time  of  exposure  to  be  tested  is  a fairly 
large  fraction  of  a second,  the  help  of  a friend,  a fair- 
sized  roll  of  white  paper,  and  a good  light  are  all  that  is 


EXPOSURE,  STOPS,  AND  SHUTTERS 


273 


required.  Place  the  friend  20  or  30  feet  in  front  of 
the  camera,  holding  the  roll  of  paper  in  one  hand  ; 
arrange  the  picture  so  that  the  shoulder  of  the  hand 
holding  the  paper  is  in  the  centre  of  the  picture  and 
the'  whole  of  the  roll  of  paper  is  visible  when  the 
longest  side  of  the  plate  is  horizontal.  If  the  paper  is 
then  whirled  round  at  arm’s  length  it  will  during  the 
whole  of  its  path  be  within  the  limits  of  the  picture. 
Then  let  the  paper  be  whirled  round  so  that  one 
revolution  is  made  in  one  second,  and  at  the  same  time 
take  a photograph  with  the  shutter  set  at  the  speed  to 
be  tested. 

It  is  not  hard  with  a little  practice  to  whirl  the  arm 
round  once  a second,  but  if  this  proves  inconvenient 
the  time  of  one  revolution  can  easily  be  measured  by 
taking  the  time  of  1 0 or  more  revolutions ; and  the 
necessary  changes  can  easily  be  introduced  into  the 
calculations. 

When  the  photograph  is  developed  the  image  of  the 
roll  of  paper  will  not  appear  sharp,  but  spread  out  into 
the  sector  of  a circle,  owing  to  the  angle  through 
which  the  paper  has  moved  during  the  exposure.  The 
angle  moved  through  can  be  measured  with  a pro- 
tractor, and  the  time  of  exposure  calculated  from  this. 

Examiile, — The  paper  is  found  to  be  whirled  round 
once  in  1*2  seconds,  and  from  the  photograph  it  is  found 
to  have  moved  through  an  angle  of  15°  during  the 
exposure. 

Here  the  roll  of  paper  revolves  through  360°  in  1*2 
seconds,  or  through  1°  in  1*2  360  seconds,  or  through 


15°  in 


15  X 1*2 

360 


seconds. 


Hence  time  of  exposure 


15  X 1*2 

360" 


1 


20 


sec. 


Hence  the  time  of  exposure  is  one-twentieth  of  a 
second. 

(2)  For  short  exposures  the  first  method  is  not 

T 


274 


PHOTOGRAPHIC  OPTICS 


reliable,  and  another  requiring  more  apparatus  must 
be  adopted.  A disc,  whirling  uniformly  at  a fair  speed, 
is  required  ; it  may  be  whirled  either  by  clock-work  or 
by  hand,  and  it  should  revolve  at  least  three  or  four' 
times  a second.  An  electromotor,  if  available,  is  very 
suitable,  for  when  the  driving  current  is  kept  constant 
the  speed  keeps  very  nearly  constant.  But  a hand 
arrangement,  with  a large  wheel  driving  a small  one  by 
means  of  a belt,  can  with  practice  be  driven  at  a very 
nearly  constant  speed. 

Cover  the  disc,  which  should  be  made  as  large  as  is 
convenient,  with  black  or  dark  paper,  and  on  this  paste 
a small  sector  of  white  paper.  First  make  an  exposure 
when  the  disc  is  at  rest,  to  show  the  actual  size  of  the 
image  of  the  sector ; then  whirl  the  disc  at  a known  rate 
and  make  an  exposure  with  the  shutter  to  be  tested. 

On  development,  the  photograph  of  the  sector  when 
still  will  be  found  to  be  sharp,  while  that  taken  when 
the  sector  was  moving  will  be  spread  out.  By  com- 
parison of  the  two  images  and  the  aid  of  a protractor 
the  angle  moved  through  by  the  disc  during  the  time 
of  exposure  can  be  found. 

The  number  of  revolutions  made  in  one  second  by 
the  disc  should  be  ascertained  directly  and  not  from 
the  speed  of  the  multiplying  wheel,  for  there  is  always 
some  slip  when  a belt  is  used ; the  most  convenient 
way  is  to  make  a small  blunt  projection  on  the  axle  or 
pulley  which  carries  the  disc,  and  to  feel  this  with  the 
finger,  ^ince  this  projection  is  near  the  axis  of  rotation, 
its  speed  is  small  and  it  will  not  hurt  the  finger ; the 
number  of  revolutions  in  a given  time,  30  seconds  for 
instance,  should  be  counted,  and  the  time  of  one 
revolution  calculated  from  this. 

Examiile. — The  disc  makes  107  revolutions  in  30 
seconds ; and  the  sector  is  found  to  move  through  an 
angle  of  23°  during  the  exposure. 

Since  the  disc  goes  through  360°  in  one  revolution 
the  exposure  was  23/360  of  the  time  taken  by  the  disc 


EXPOSURE,  STOPS,  AND  SHUTTERS 


275 


to  revolve  once.  The  disc  revolves  107  times  in  30 
seconds,  hence  it  revolves  once  in  30/107  seconds; 

30  23 

therefore  time  of  exposure  = X 2“^=  '02 

second  (nearly),  or  the  exposure  was  about  one-fiftieth 
of  a second. 

In  experimental  work  the  light  of  an  electric  spark 
has  been  used  instead  of  a shutter ; the  exposure  is 
very  much  less  than  that  possible  with  the  quickest 
shutter.  In  this  case  the  revolving  disc  can  still  be 
used,  but  it  must  be  revolved  at  a much  greater 
speed,  and  the  time  of  revolution  must  be  estimated  by 
tlie  aid  of  a tuning-fork. 

131.  Efficiency  of  a Shutter. — When  using  a shutter 
the  effect  on  the  plate  is  not  correctly  measured  by  the 
total  interval  that  elapses  between  the  time  when  the 
shutter  begins  to  open  and  that  when  it  is  finally  shut, 
or  in  other  words  we  must  not  reckon  the  exposure  as 
if  the  total  aperture  were  unclosed  from  the  beginning 
to  the  end  of  the  exposure.  A little  reflection  will  show 
that  the  exposure  may  roughly  be  divided  into  three 
periods,  the  first  that  during  which  the  shutter  is 
opening,  the  second  that  during  which  the  full  aperture 
is  open,  the  third  that  during  which  the  shutter  is 
closing ; with  some  shutters  the  second  period  is  absent. 
During  the  first  and  third  periods  the  whole  aperture  is 
not  unclosed,  and  consequently  the  illumination  of  the 
plate  is  not  then  as  great  as  during  the  time  when  the 
aperture  is  quite  open. 

What  we  want  to  know,  for  practical  purposes,  is 
what  exposure,  with  the  whole  aperture  unclosed,  is 
equivalent  to  that  actually  given ; we  shall  call  this 
the  equivalent  exposure,  and  the  time  between  the  first 
opening  and  the  last  closing,  the  nominal  exposure. 

equivalent  exposure  . n i . 

I he  ratio  — ^ ^ is  called  the  efficiency 

nominal  exposure 

of  a shutter. 

Let  us  take  the  simplest  case  possible,  though  it  is 


276 


PHOTOGRAPHIC  OPTICS 


not  one  realized  in  practice  ; imagine  the  aperture  to 
be  square,  and  let  this  be  uncovered  and  covered  by  the 
sliding  in  front  of  it,  with  uniform  speedy  of  a panel 
pierced  with  a square  hole  equal  in  size  to  the  aperture  ; 
the  hole  being  so  arranged  as  to  exactly  coincide  with 
the  aperture  in  one  position.  Consideration  will  show 
that  every  part  of  the  aperture  is  uncovered  for  a time 
equal  to  half  the  time  of  the  total  exposure,  and  the 
aperture  is  of  the  same  breadth  from  top  to  bottom ; 
hence  the  total  exposure  is  the  same  as  would  have 
been  given  with  the  whole  aperture  uncovered  for  half 
the  time.  Thus  the  efficiency  is  one-half. 

We  have  here  assumed  the  speed  of  the  shutter  to 
be  uniform ; this  is  not  at  all  likely  to  be  the  case,  and 
except  when  the  speed  is  uniform  the  efficiency  is  very 
hard  to  calculate,  even  in  simple  cases  when  the 
shutter  falls  under  gravity  ; and  in  most  cases  the  mode 
of  motion  is  unknown.  But  a consideration  of  the 
efficiency  of  different  forms  of  shutters  with  uniform 
speed  is  instructive,  as  it  gives  some  rough  indication 
of  the  efficiency  in  practice. 

132.  Efficiencies  at  Uniform  Speed. — The  forms  of 
shutters  considered  are  shown  in  Fig.  85 ; the  calculations 
are  best  made  by  the  aid  of  the  Integral  Calculus,  and  are 
very  troublesome  by  elementary  methods,  hence  the 
results  only  are  stated ; the  verification  of  the  results 
will  provide  an  exercise  for  mathematical  readers.  An 
approximate  estimate  can  be  made  in  some  cases  by  a 
method  to  be  shown  in  the  next  section.  The  aperture 
is  taken  to  be  circular  in  each  case. 

(a)  When  the  sliding  panel  is  square,  so  that  the 
opening  begins  from  one  side,  the  efficiency  is  ‘500. 

(b)  When  the  edges  of  the  sliding  panels  are  straight 
and  there  are  two  of  them  opening  from  the  centre,  as 
drawn,  and  closing  again  to  the  centre,  the  efficiency 
is  *576. 

(c)  When  there  is  a single  sliding  panel  with  a 
circular  hole,  the  efficiency  is  *424. 


EXPOSUKE,  STOPS,  AND  SHUTTEPtS 


277 


278 


PHOTOGRAPHIC  OPTICS 


{d)  When  there  are  two  sliding  panels  each  with  a 
circular  hole,  opening  and  closing  at  the  centre,  the 
efficiency  is  '424,  the  same  as  in  the  last  case. 

(e)  When  the  exposure  is  made  by  the  rise  and  fall  of 
a panel,  with  its  end  circular  and  equal  to  the  radius  of 
the  aperture,  the  efficiency  is  '576,  the  same  as  for  (6). 

if)  When  there  are  two  panels  with  circular  holes 
which  revolve  about  a pivot  o ; let  ;2:  be  the  ratio  of  the 
radius  of  the  aperture,  to  the  distance  of  the  pivot  from 
the  centre  of  the  aperture. 

The  efficiency  can  be  calculated  for  simple  cases 
when  z = 0 the  efficiency  — '424 

„ «=l/2„  „ =.-422 

» = ’297 

When  z = 0 the  pivot  is  at  an  infinite  distance  and 
the  motion  of  the  panels  becomes  sliding  motion  as  in 
{/) ; the  result  as  we  should  expect  is  the  same  in  the 
two  cases. 

{(j)  When  the  aperture  is  uncovered  by  two  panels 
turning  about  a pivot  fixed  outside  the  aperture,  if 
have  the  same  meaning  as  in  the  last  case,  then 

when  z = 0 the  efficiency  = *576 

„ « = 1/2  „ „ = -678 

„ z = I „ „ = -703 

In  both  (/)  and  (g)  it  should  be  noticed  that  z = 1 
means  that  the  pivot  is  in  the  circumference  of  the 
aperture. 

133.  Calculation  of  Efficiency  in  General. — The 

motion  of  a shutter  is  in  general  not  uniform  as  assuihed 
in  the  last  article,  but  a method  is  given  below  (§  134) 
by  means  of  which  the  motion  can  be  found,  and  from 
this  an  estimate  of  the  efficiency  can  be  made. 

The  quantity  of  light  which  during  any  short  interval 
falls  on  the  centre  of  the  plate  is  proportional  to  the 
product  of  the  interval  into  the  average  area  uncovered 
during  that  interval ; we  can  therefore  make  an  estimate 
of  the  total  effect  of  the  exposure,  by  dividing  up  the 


EXPOSURE,  STOPS,  AND  SHUTTERS 


279 


whole  time  of  the  exposure  into  short  intervals,  finding 
the  average  areas  uncovered  during  each  interval,  and 
adding  together  the  product  of  those  areas  into  the 
corresponding  intervals.  If  the  quantity  so  found  be 
divided  by  the  total  area  the  result  will  be  the  equiva- 
lent exposure. 

Thus  let  T be  the  nominal  exposure,  let  this  be 
divided  into  a number  of  small  equal  intervals  each 
equal  to  and  let  the  average  areas  uncovered  during 
each  interval  be,  a^^  <^3,  etc.,  and  let  A be  the  total 
area,  then 

t + a.^  t + Uo  t + etc. 
equivalent  exposure'  = — — — — 

, ,,  • a.  t + a.^  t ao  t etc. 

and  the  emciency  = ^ — ^7T  


Examq)le, — The  nominal  exposure  was  1/10  second, 
for  sec.  yy  of  the  aperture  was  uncovered 

i_  "i_ 

100  10 

1 y 

1 0 0 2 


JJ  10  0 


10  0 

. 

5)  10  0 


the  whole  aperture  was  uncovered 
I of  the  aperture  was  uncovered 


>>  100  5)  20  55  ))  J) 

Hence  reckoning  as  above  the  efficiency  is 

(lIF  TF  d"  ■2 ) TFTT  d“  TFF  (nF  d"  iV  d"  . 


= •53 


TF 

and  the  equivalent  exposure  is  *053  sec. 

The  following  table  (p.  280)  has  been  calculated  to 
facilitate  the  calculation  of  efficiency ; it  shows  the 
fraction  of  the  whole  aperture  that  is  unclosed  at  each 
tenth  of  the  movement  of  the  panel  or  panels  required  to 
fully  unclose  the  aperture. 

If  other  values  besides  these  given  are  required  the 
given  values  should  be  plotted  on  squared  paper  and  the 
points  found  joined  by  a continuous  curve  in  the  manner 
familiar  to  engineers ; intermediate  values  may  then  be 
read  off,  approximately,  from  the  diagram. 


280 


rnoTOGRArHic  optics 


Table  showing  Fraction  of  Aperture  Unclosed  for  each 
Tenth  of  Movement  of  Panel 


Fraction  of  dis- 
tance moved  by 
panel 

•1 

• 2 

•3 

•4 

*5 

•6 

•7 

•8 

•9 

1-0 

Fraction  of  area  unclosed 
for  cases  shown. 

See  Fig.  85. 

a 

h,  e 

•0528 

•1395 

1 

•2500 

1 

•3738 

•5000 

•0-202 

•7500 

•8005  1 

•9472 

1-0000 

•1272 

•2524 

1 

•3774 

•5000 

•0090 

•7210 

•8120 

i 

•8944 

•9030 

1-0000 

c,  d 

•0370 

•1050 

•lt80 

1 

•2790 

•3910 

•5000 

•0220 

•7470 

•8728 

1 -0000 

134.  Experimental  Examination  of  Shutters. — Cap- 
tain Abney  has  devised  a method  for  examining  the 
motion  of  shutters  by  means  of  which  a diagram  is  drawn 
showing  the  position  of  the  parts  at  any  instant  during 
the  exposure. 

The  method  is  to  place  in  the  aperture  of  the  shutter  a 
piece  of  cardboard,  in  the  middle  of  which  is  cut  a slit 
at  right  angles  to  the  direction  of  motion  of  the  shutter ; 
the  image  of  this  slit  is  thrown  on  a plate  or  film,  which 
is  moving  in  a direction  at  right  angles  to  the  slit.  If 
the  plate  were  at  rest  the  effect  of  the  exposure  would 
be  to  produce  a photograph  of  the  slit,  but  when  the 
plate  moves  this  is  stretched  out  into  a band ; the 
breadth  of  this  band  at  any  part  of  the  exposure  shows 
the  breadth  of  the  opening  of  the  shutter.  If  a scale  of 
times  can  be  marked  on  the  diagram  we  can  by  inspec- 
tion find  the  state  of  the  shutter  at  any  time  required. 

The  shutter  aperture  with  slit  is  shown  in  Fig.  86,  in 
which  C C are  the  pieces  of  card  containing  the  slit,  and 
S S are  the  moving  panels  of  the  shutter,  shown  partly 
withdrawn. 

The  plate  can  be  moved  by  hand,  but  this  is  not  very 
convenient  as  it  requires  the  use  of  a dark  room  ; the 
more  convenient  arrangement  is  to  roll  a flexible  sensi- 


EXPOSURE,  STOPS,  AND  SHUTTERS 


281 


tive  film  round  a drum  which  is  made  to  revolve  rapidly  ; 
this  arrangement  is  shown  in  Fig.  87,  in  which  the  drum 
is  arranged  in  a box  made  to  fit  the  camera  like  a dark 
slide.  The  spindle  and  small  pulley  at  the  side  are  for 
driving  the  drum ; the  most  convenient  thing  for  this 


purpose  is  an  electromotor  which  can  be  made  to  run  at 
a very  nearly  constant  rate. 

The  general  arrangement  of  the  apparatus  is  shown  in 
Fig.  88  ; on  the  extreme  left  is  an  electric  arc  lamp 
which  provides  the  necessary  light : next  is  placed  a lens 
which  acts  like  the  condenser  of  a lantern,  throwing  a 
beam  of  light  on  the  shutter  : next  comes  the  lens  to  be 
tested,  fitted  with  the  cardboard  slit  (which  is  here 


282 


PHOTOGRAPHIC  OPTICS 


horizontal)  : next  comes  a wheel,  to  be  explained  below, 
for  timing  purposes  : next  is  the  camera  with  the  box 
containing  the  revolving  drum : at  the  end  is  the 
electromotor  to  drive  the  drum. 

The  wheel  is  so  placed  that  the  image  of  the  slit  can 
be  obscured  by  the  spokes ; if  the  wheel  revolves  at  a 
definite  rate  the  light  from  the  slit  will  be  shut  off  at 
definite  intervals.  Lines  are  thus  marked  across  the 


Fig.  87. 


diagram,  the  distances  between  them  representing  equal 
intervals  of  time. 

The  holes  round  the  rim  are  for  measuring  tlie  speed 
of  the  wheel,  air  is  blown  through  them  as  the  wheel 
revolves,  forming  a syren ; the  pitch  of  the  note  can 
be  found  by  comparison  with  a tuning-fork  or  other 
instrument  of  known  pitch,  and  thus  the  number  of 
holes  which  pass  the  air-jet  in  one  second  is  known, 
and  the  speed  of  revolution  can  be  reckoned. 

For  a fuller  explanation  of  this  process  we  must  refer 
readers  to  some  text-book  on  sound  where  the  formation 


EXPOSURE,  STOPS,  AND  SHUTTERS 


•283 


284 


PHOTOGRAPHIC  OPTICS 


of  notes  of  definite  pitch  and  the  action  of  the  syren  are 
explained ; the  following  table  gives  the  number  of 
vibrations  per  second  required  to  produce  the  notes  of 
an  octave  beginning  at  middle  C. 


Scientific  scale. 

Society  of  Arts 

c 

512  ... 

528 

C sharp 

540  ... 

559 

D 

576  ... 

594 

D sharp 

600  ... 

622 

E 

640  ... 

660 

F 

683  ... 

704 

F sharp 

720  ... 

745 

Gr  ... 

768  ... 

792 

G sharp 

800  ... 

837 

A 

853  ... 

880 

A sharp 

900  ... 

932 

B 

960  ... 

990 

C 

...  1024  ... 

,..  1056 

135.  The  Study  of  a Shutter  Diagram.  — Let  us 

examine  the  diagram  shown  in  Fig.  89,  which  is  given 
by  Abney.  Here  the  direction  of  motion  of  the  film 
was  parallel  to  A B,  and  the  slit  was  at  right  angles  to 
this  direction ; the  white  lines  across  are  due  to  the 
interruptions  caused  by  the  spokes  of  the  revolving 
wheel. 

The  first  thing  to  notice  is  that  the  interruptions  are 
marked  at  equal  distances  along  the  film,  showing  that 
the  drum  revolved  uniformly.  Since  the  line  A E B 
remains  straight  for  a long  time  it  is  clear  that  the 
shutter  was  one  in  which  tlie  opening  began  at  one  end 
of  the  slit,  as  in  Fig.  85  (A) ; the  sliding  of  the  panel  is 
shown  by  the  sloping  line  A C,  and  the  full  aperture  is 
reached  at  E C.  The  portion  between  E C and  B F 
represents  the  interval  during  which  the  aperture  was 
fully  unclosed ; the  straight  line  C D and  the  sloping 


EXPOSURE,  STOPS,  AND  SHUTTERS 


285 


line  B D show  that  the  aperture  was  closed  by  a panel 
sliding  over  it  in  the  same  direction  as  the  former  one. 

The  diagram  could  have  been  given  by  a shutter  like 
that  in  Fig.  85  (yl),  if  the  aperture  in  the  sliding  panel 
instead  of  being  square  were  a rectangle,  so  that  the 
aperture  may  remain  fully  uncovered  for  a finite  time. 

The  line  A C is  straight,  which  shows  that  the  speed 
of  the  opening  panel  was  uniform,  but  B D is  curved  and 
convex  towards  C D,  showing  that  the  motion  of  the 
closing  panel  was  retarded.  To  sum  up,  the  opening 
took  2 1 intervals,  the  aperture  was  fully  open  for  3^ 
intervals,  and  the  closing  took  about  3|  intervals.  In 
this  particular  case  the  note  given  by  the  syren  was  E, 
which  means  that  640  holes  passed  the  air-jet  in  one 
second  ; the  wheel  used  had  6 spokes  and  3 6 holes  in 

A E B 

) C F D 

; Fig.  89. 

the  rim,  or  6 holes  to  each  spoke,  which  makes  the 
number  of  eclipses  by  the  spokes  to  be  640/6,  or  107 
nearly  in  one  second ; thus  the  time  between  the 
successive  eclipses  is  1/107  = *0093  sec.,  or  roughly  one 
hundredth  of  a second.  Calculating  from  this  we  find 
that  the  total  time  of  exposure  was  *095  sec.,  the  open- 
ing took  *023  sec.,  the  aperture  was  fully  open  for  *032 
sec.,  and  the  closing  occupied  *04  sec.  ; the  times  are 
given  to  thousandths  of  a second,  but  it  is  not  likely  that 
the  method  is  accurate  to  this  extent ; it  would  be  safer 
to  give  the  results  to  the  nearest  hundredth  of  a second. 

To  calculate  the  efficiency  we  must  remember  that 
for  this  kind  of  shutter  the  value  given  in  § 132  (a) 
was  *5  ; so  if  we  regard  the  opening  and  shutting  as 


286 


PHOTOGRAPHIC  OPTICS 


uniform,  the  time  with  full  aperture  to  which  the  time 
of  opening  and  shutting  is  equivalent,  is  half  the  actual 


Fig.  90  {a). 


Fig.  90  {b). 

time  taken  to  open  and  close,  that  is,  to  half  the  sum  of 
2|-  and  3|,  or  of  6;^  intervals ; hence  adding  in  the 


EXPOSURE,  STOPS,  AND  SHUTTERS 


287 


time  of  full  aperture,  equivalent  exposure  = 3|-  + 3|- 
= 6f  intervals,  but  normal  exposure  = lOj  intervals, 
hence  dividing  the  equivalent  by  the  nominal  exposure 
we  get  for  the  efficiency  ‘64. 

^ 136.  Timing  by  the  Speed  of  the  Drum. — After 


Fig.  90  (c). 


Fig.  90  (d). 


several  experiments  Abney  found  that  the  motion  of 
the  drum  kept  so  nearly  uniform  that  the  time  of 
exposure  could  be  estimated  from  its  speed  of  rotation 
and  from  its  diameter.  To  find  the  speed  of  rotation 
use  was  made  of  an  old  turnstile  counter,  which  was 


288 


rnOTOGRAPHIC  OPTICS 


attached  to  the  axle,  and  the  number  of  revolutions  in 
one  minute  were  counted. 

Exam'ple. — It  is  found  that  the  drum  makes  400 
revolutions  in  one  minute,  and  its  diameter  is  2 inches, 
the  total  length  of  the  shutter  diagram  is  inches ; 
find  the  nominal  exposure. 

The  radius  is  1 inch,  hence  the  circumference  is  2 tt 
inches,  so  that  the  film  moves  through  a distance  of 
400  X 27r  inches  in  a minute,  or  400  X 27r/60  inches  in 
one  second. 

During  the  exposure  the  film  moves  through  4|^ 
inches,  hence  the  duration  of  the  exposure  is 
6 0 

X 4^^-  = *096  sec.  about,  or  nearly  one-tenth 

400  X 27r 
of  a second. 

In  Fig.  90  are  shown  some  of  Abney’s  diagrams ; 

(a)  is  that  for  a drop  shutter  (remember  that  the 
image  of  the  slit  is  inverted,  so  that  the  bottom 
of  the  diagram  corresponds  to  the  top  of  the 
shutter). 

(5)  contains  diagrams  for  Thornton  and  Picard’s 
shutter. 

(c)  is  a diagram  for  Hawkins’  shutter. 

{d)  contains  diagrams  for  a Key  shutter. 

137.  Unequal  Exposures  at  Different  Parts  of  the 
Plate. — It  is  a matter  of  common  experience  that  when 
a shutter  is  used  the  edges  of  the  plate  are  often  less 
exposed  than  the  centre,  to  the  detriment  of  the 
picture  ; a shutter  diagram  enables  us  to  study  the 
exposures  of  the  different  portions  of  the  plate.  To 
prevent  confusion  it  should  be  remembered  that  we 
have  reckoned  the  efficiencies  only  for  the  centre  of  the 
plate. 

Let  Fig.  91  represent  a section  through  the  axis 
of  the  lens;  let  the  lens  be  as  shown,  and  D E the 
section  of  the  full  aperture  of  the  shutter  in  its 
proper  position.  Consider  an  oblique  pencil,  whose 
extreme  rays  cut  the  line  D E at  the  point  a a ; this. 


EXPOSURE,  STOPS,  AND  SHUTTERS 


289 


pencil  falls  on  the  lower  part  of  the  plate  ; let  a central 
pencil  cut  the  line  D E at  6 5,  and  an  oblique  pencil  on 
the  lower  side  of  the  axis,  in  c c.  Suppose  for  example 
that  the  shutter  opens  arid  closes  at  the  centre,  then  it 
is  evident  that  the  pencil  h h will  be  admitted  first,  and 
the  pencils  aa^  cc  completely  when  the  sliding  panels 
are  above  the  highest  a and  below  the  lowest  c ; also 
when  the  panels  close  the  oblique  pencils  are  cut  off 
first.  On  both  these  accounts  the  oblique  pencils  are 


Fir,,  91. 


admitted  for  a considerably  shorter  time  than  the 
central  one. 

Now  take  the  case  of  a shutter  whose  diagram  is 
that  of  Fig.  90  {a),  opening  at  one  side,  and  let  the 
diagram  be  placed  on  the  section,  as  shown,  so  that  the 
top  and  bottom  lines  S S pass  through  D and  E.  To 
understand  the  meaning  of  the  diagram,  it  should  be 
remembered  that  when  it  was  made  the  film  moved  in 
the  direction  S S,  and  consequently  the  breadth  of  the 
opening  of  the  shutter  at  any  instant  is  given  by  the 

U 


290 


PHOTOGRAPHIC  OPTICS 


breadth  of  the  diagram  measured  perpendicular  to  S S, 
and  at  a point  corresponding  to  the  given  instant. 

From  the  points  a a,  hh^  cc  draw  parallels  to  S S to 
cut  the  diagram ; the  darkly  shaded  parts  of  the  diagram 
show  the  areas  between  these  pairs  of  lines.  To  avoid 
mistake  it  must  be  remarked  that  the  diagram  shows 
the  nature  of  motion  of  the  shutter,  but  that  the 
motion  really  takes  place  along  D E.  The  shutter 
opens  from  one  side  as  the  lower  horizontal  line  S S 
shows ; while  the  panel  moves  across  c c,  as  shown  by 
the  left-hand  edge  of  the  lowest  dark  portion,  the  lower 
oblique  pencil  is  being  admitted  ; then  in  succession 
the  lower  oblique  pencil  is  admitted  : the  shutter 
remains  fully  open  for  a short  time,  and  then  another 
panel  moves  in  the  same  direction  as  the  former, 
cutting  off  the  pencils  in  the  order  of  their  admission. 

Let  us  now  fix  our  attention  on  the  pencil  a a and 
trace  its  history : this  pencil  is  admitted  last ; its 
admission  begins  when  the  panel  reaches  the  position 
shown  by  the  lower  a on  the  left ; the  pencil  is  fully 
admitted  when  the  upper  a on  the  same  side  is  reached  : 
the  pencil  continues  to  be  fully  admitted  till  the  closing 
panel  reaches  the  position  shown  by  the  lower  a on  the 
right,  and  is  completely  cut  off  when  the  upper  a on 
the  same  side  is  reached.  We  see  that  the  time  during 
which  the  pencil  is  fully  admitted  is  given  by  the 
horizontal  distance  (parallel  to  S S),  between  the  upper 
a on  the  left  and  the  lower  a on  the  right ; vertical 
lines  have  been  drawn  through  these  points  and  joined 
by  a horizontal  line  just  below  the  diagram.  Similar 
remarks  apply  to  the  pencils  h h and  c c,  and  similar 
constructions  have  been  made  to  show  the  duration  of 
the  times  of  complete  admission. 

It  is  worth  noticing  that  the  portion  a a a a of  the 
shutter  diagram  may  be  regarded  by  itself  as  the  com- 
plete diagram  for  the  pencil  a and  the  efficiency  for 
the  pencil  might  be  calculated  from  it ; similarly  for 
the  pencil  c c.  It  should  also  be  noticed  that  the  portion 


EXPOSURE,  STOPS,  AND  SHUTTERS 


291 


6 5 5 & is  all  that  applies  to  the  central  pencil,  so  that  the 
efficiency  calculated  from  the  whole  diagram  would  not 
be  correct  ; it  is  evident  from  this  that  with  a small 
stop  and  the  shutter  at  a considerable  distance  from 
the  lens,  the  whole  of  the  diagram  may  not  apply  to 
the  central  pencils.  In  this  case  we  see  that  the  times 
of  total  admission  of  the  three  pencils  are  nearly  equal, 
and  also,  from  the  great  similarity  of  the  portions  aa  a a, 
hh  hh,  G c Gc,  that  the  efficiencies  for  each  portion 
are  nearly  equal ; but  the  exposures  do  not  take  place 
quite  simultaneously. 

We  have  here  considered  the  circumstances  of 
portions  of  the  plate  in  a line  parallel  to  the  slit, 
placed  in  the  centre.  Our  conclusions  will  not  in  general 
hold  good  for  portions  in  a line  at  right  angles  to  this. 


Fig.  92. 


through  the  centre  of  the  plate.  This  latter  case  can 
be  investigated  by  taking  a diagram  with  the  slit  in 
the  shutter  at  right  angles  to  its  former  position ; the 
discussion  of  the  results  will  be  similar  to  that  given. 
A pair  of  diagrams,  taken  by  Abney,  with  the  slit  in 
two  directions  at  right  angle,  are  given  in  Fig.  92. 

138.  Focal  Plane  Shutter. — This  shutter  differs 
widely  from  any  of  those  we  have  considered,  for  these 
are  either  very  near  to  the  lens,  or  are  at  the  dia- 
phragm, while  the  focal  plane  shutter  is  close  to  the  plate. 
The  exposure  is  given  by  the  rapid  passage  of  a narrow 
slit  across  the  plate  ; this  clearly  lends  itself  to  rapid 
exposures,  which  can  be  adjusted  by  altering  the 
breadth  of  the  slit  (Fig.  93). 

This  shutter  has  however  the  disadvantage  of  ex- 


292 


PHOTOGRAPHIC  OPTICS 


posing  the  different  parts  of  the  plate  at  different 
times,  the  difference  betvv^een  the  times  at  which  the 
two  extremities  are  exposed  being  in  most  cases  greater 
than  the  time  of  exposure ; this  produces  a distorting 
effect  on  the  picture  of  a moving  object.  Suppose  for 
instance  that  the  mast  of  a rapidly  moving  ship  is 
being  photographed,  and  that  the  shutter  slit  travels 
from  the  bottom  of  the  plate  to  the  top.  Then,  allow- 
ing for  the  reversal  of  the  picture  on  the  plate,  the 
image  of  the  top  of  the  mast  will  be  admitted  first,  and 
that  of  the  bottom  of  the  mast  after  an  interval  three  or 
four  times  the  lengtli  of  the  exposure,  and  during  this 


Fig.  93. 

interval  the  ship  will  have  moved ; the  intermediate 
portions  will  be  exposed  at  intermediate  times.  In  the 
photograph  the  base  of  the  mast  will  be  in  advance  of 
the  top,  and  the  mast  will  appear  to  slope  backwards ; 
in  this  particular  case  the  sloping  backwards  may 
improve  the  picture,  by  giving  the  ship  a rakish  appear- 
ance, but  it  is  not  truthful.  Besides  this,  cases,  such 
as  a man  walking  rapidly,  can  be  imagined  in  which 
the  effect  would  be  disastrous. 

139. — We  have  considered  some  typical  forms  of 
shutters,  but  there  are  many  others  which  are  to  be 
found  in  makers’  catalogues  and  photographic  annuals  ; 


EXPOSUKE,  STOPS,  AND  SHUTTEPS 


293 


the  reader  who  is  interested  in  the  matter  may  also 
refer  to  the  Traite  Encydopedique  de  Photographies'^ 
vol.  i.  pp.  150 — 205,  where  much  interesting  inform- 
ation is  given. 

^ From  the  discussion  of  the  question,  it  appears  that 
an  exposure  made  with  a shutter  is  not  so  satisfactory 
as  a slow  one  made  by  hand,  and  that  in  order  to 
secure  equal  exposure  of  all  parts  of  the  plate,  the  form 
shown  in  Fig.  85  {A)  is  preferable  to  one  like  Fig.  85 
{B)  or  {D)s  which  open  in  the  centre. 

^ By  C.  Fabre.  Gauthier- Villars,  Paris,  1890. 


CHAPTER  YII 


ENLARGEMENT,  REDUCTION,  DEPTH  OF  FOCUS,  AND 
HALATION 

140.  Introductory. — The  difficulties  of  making  large 
photographs  directly  are  very  great ; the  apparatus 
required  is  large,  heavy,  and  inconvenient  to  carry 
about ; a lens  of  large  diameter  must  be  employed, 
which  is  troublesome  to  make,  and  in  consequence  is 
expensive ; the  plates  are,  from  their  size,  difficult  to 
manipulate  ; and,  lastly,  the  expenditure  in  materials  is 
considerable,  for  large  plates  are  expensive,  and  need 
large  quantities  of  chemicals. 

It  is,  therefore,  much  more  convenient  in  many  cases 
to  take  photographs,  first  on  a small  scale,  and  then,  if 
the  negatives  are  good,  to  make  enlargements ; a great 
saving  results,  both  in  the  original  cost  of  the  apparatus 
and  also  in  the  current  expenses,  for  the  failure  of  a 
small  negative  is  not  so  costly  as  that  of  a large  one. 
It  is  not  surprising,  therefore,  that  enlarging  is  very 
popular.  Reduction  is  required  mainly  to  make  lantern 
slides  from  negatives  which  are  too  large  to  admit  of 
contact  printing. 

Many  forms  of  enlarging  and  reducing  apparatus  are 
sold,  differing  little  in  principle,  but  many  of  them  are 
of  needless  complexity,  and  most  of  them  are  very  highly 
priced. 

It  is  proposed  first  to  give  some  account  of  the 
optical  principles  of  enlarging,  and  then  to  show  how 
those  who  cannot  afford  expensive  apparatus  can  pro- 

294 


295 


ENi^ARGEMENT,  REDUCTION,  ETC. 


duce  enlargements  at  a comparatively  small  cost  by  the 
aid  of  an  ordinary  camera  and  lens. 

141.  Optical  Principles.  — The  general  principles 
which  apply  to  the  production  of  pictures  of  a size 
different  from  that  of  the  original  have  been  explained 
in  § 37  ; this  section  applies  equally  to  reduction. 

The  essential  parts  of  the  enlarging  or  reducing 
apparatus  are,  the  negative,  the  lens,  the  sensitive  plate 
or  paper  which  receives  the  image,  and  the  source  of 
light.  The  light  shines  through  the  negative,  and  an 
enlarged  or  reduced  picture  of  it  is  formed  by  the  lens 
on  the  paper  or  plate,  and  results  in  the  formation  of  a 
positive  picture. 

Thus,  in  any  enlarging  or  reducing  apparatus,  the  re- 
quirements are,  uniform  illumination  of  the  negative, 
and  facilities  for  adjusting  the  relative  distances  of 
negative,  lens,  and  sensitive  receiving  surface. 

Similar  remarks  apply  to  the  optical  lantern,  with 
the  exception  that,  instead  of  a negative  a positive  is 
used,  and  a positive  picture  is  thrown  on  the  screen. 

The  illumination  may  be  either  direct  sunlight, 
diffused  daylight,  or  artificial  light,  provided  the  light  is 
uniformly  distributed  over  the  negative.  To  distribute 
the  light,  a lens,  called  a condensing  lens,  is  generally 
used,  though  this  can  be  in  some  cases  dispensed  with. 

We  shall  now  study  the  action  of  the  condensing  lens, 
and  for  this  purpose  shall  consider  Wood  ward’s  apparatus, 
which  was  one  of  the  earliest  pieces  of  apparatus  for 
enlarging. 

141a.  Woodward^s  Apparatus. — In  the  early  days  of 
photography,  before  the  introduction  of  the  very  sensitive 
plates  and  papers  we  now  employ,  the  production  of  an 
enlargement  was  a matter  of  difficulty.  When  enlarg- 
ing on  slow  silver  paper,  in  order  to  reduce  the  time  of 
exposure  within  reasonable  limits,  it  was  necessary  not 
only  to  use  direct  sunlight,  but  also  to  concentrate  it  as 
much  as  possible  by  means  of  a condensing  lens. 

' An  early  apparatus  was  that  of  Woodward,  shown  in 


296 


PHOTOGKAPHiC  OPTICS 


Fig.  94. 


ENLARGEMENT,  REDUCTION,  ETC. 


297 


Fig.  94  ; here  A B is  a mirror  to  reflect  the  sunlight 
on  to  the  condensing  lens  I ; its  position  was  regulated 
by  means  of  the  screws  indicated  at  B and  C.  The 
lens  I concentrates  the  light  to  a point  y*near  the  objec- 
tive L,  so  that  all  the  light  falls  on  L,  and  none  is 
interrupted  by  the  mounting.  J is  the  negative, 
carried  in  a holder  which  can  be  moved  backwards  or 
forwards,  and  fixed  by  means  of  the  screw  K;  the  enlarged 
picture  is  projected  by  the  lens  L on  a screen  in  front, 
not  shown  in  the  picture.  The  whole  arrangement  is 
carried  by  a wooden  box  E F G H,  and  is  placed  in  a 
window  so  that  the  box  passes  through  a hole  in  the 
shutter,  the  part  A B being  outside,  and  carefully 
fitted  so  that  no  light  can  enter  the  room  except 
through  the  lens. 

There  was  at  one  time  a lively  discussion  about  this 
and  other  arrangements,  which  has  now  lost  most  of  its 
interest  for  us ; it  will  be  found  at  length  in  Monkhoven’s 
Optics ; we  shall  here  consider  the  theory  only  so  far  as 
it  throws  light  on  the  action  of  modern  apparatus. 

Let  us  examine  the  manner  in  which  the  enlarged 
image  is  formed  ; it  may  be  looked  at  from  two  difierent 
points  of  view. 

I.  Imagine  first  that  the  negative  is  removed,  and 
that  the  light  from  I converges  to  the  nodal  point  of 
incidence  of  the  lens  L,  then  (§  44)  the  rays  will  pass 
undeviated,  and  on  the  screen  behind  will  be  formed  a 
circle  of  light  whose  size  depends  on  the  distance  of  the 
screen  from  L.  If  now  a grating,  such  as  a piece  of  per- 
forated zinc,  be  placed  anywhere  between  I and  L,  it  will 
cut  off  some  of  the  rays,  and  its  shadow  will  constitute 
an  image  on  the  screen.  From  this  point  of  view  the 
position  of  the  object  which  intercepts  the  light  is 
immaterial. 

Even  if  the  light  from  I does  not  converge  to  the 
nodal  point  of  incidence  to  each  ray  in  the  incident  cone, 
there  will  correspond  one  particular  ray  in  the  emergent 
cone,  and  a shadow  image  will  result  as  before. 


298 


PHOTOGRAPHIC  OPTICS 


This  depends  on  the  supposition  that  one  ray  of  light, 
and  one  only,  passes  through  each  point  of  the  negative, 
so  that  a point  in  the  negative  does  not  send  out  a 
pencil  of  rays  which  can  be  converged  by  the  lens  L to 
a conjugate  focus. 

II.  On  the  other  hand,  if  any  scattering  takes  place 
when  light  passes  through  the  negative,  each  point  of 
the  negative  that  is  not  opaque  will  send  out  a small 
pencil  of  light,  which  will  fall  on  the  objective  and  be 
regularly  refracted,  and  thus  a regular  image  of  the 
negative  will  be  formed. 

We  should  thus  expect  images  of  two  different  kinds  ; 
the  first  kind  of  image  is  a shadow  image,  formed  at  all 
distances,  and  the  second  an  image  by  the  regular 
refraction  of  small  pencils  coming  from  the  different 
points  of  the  negative  ; the  best  result  evidently  will  be 
obtained  when  the  screen  is  placed  to  receive  the  second 
image,  for  the  two  images  will  then  coincide. 

The  supposition  that  one  ray  of  light  only  passes 
through  each  point  of  the  negative  is  never  strictly 
true ; even  with  the  sun  the  incident  rays  are  not 
quite  all  parallel,  for  the  sun  has  a diameter  which  sub- 
tends at  the  earth  an  angle  of  half  a degree,  and  hence 
every  point  on  which  sunlight  falls  receives  a pencil  of 
rays  whose  angle  is  half  a degree.  If  a cloud  passes 
over  the  sun  so  that  the  light  is  all  diffused,  the  effect 
becomes  very  marked.  Also,  if  artificial  light  is  used, 
the  source  of  light  is  always  of  finite  size,  and  each  point 
of  the  negative  receives  light  from  all  points  of  the 
source,  all  the  rays  received  forming  a pencil.  On  this 
account,  then,  as  well  as  because  of  scattering,  every 
clear  point  of  the  negative  sends  a pencil  of  rays  to  the 
lens,  and  a regular  image  is  formed. 

It  is  not  hard  to  see  that,  if  the  condensing  lens 
exhibits  any  spherical  aberration,  it  will  help  the  form- 
ation of  the  regular  image. 

The  conclusion  to  which  we  come  is,  that  owing  to 
the  finite  size  of  the  source  of  light,  to  the  scattering  of 


ENLARGEMENT,  REDUCTION,  ETC. 


299 


light  at  the  negative,  and  to  the  spherical  aberration  of 
the  condenser,  the  objective  forms  a regular  image,  and 
its  action  is,  on  the  whole,  the  same  as  if  it  were  form- 
ing an  image  of  an  object  at  the  same  distance  as  the 
negative  illuminated  by  diffuse  light. 

Similar  remarks  will  clearly  apply  to  the  optical 
lantern. 

142.  Concentration  of  the  Light. — In  Woodward’s 
apparatus  the  condensing  lens  I is  arranged  to  produce 
an  intense  illumination  of  the  negative  J ; it  can  be 
seen  from  the  figure  that  the  light  which  falls  on  the 
whole  area  of  I (which  is  larger  than  that  of  J)  falls  on 
the  negative  J,  and  so  by  increasing  the  size  of  the 
condenser,  we  can  increase  the  quantity  of  light  which 
falls  on  J.  The  necessity  of  intense  illumination,  when 
using  silver  paper,  led  to  the  employment  of  very  large 
condensers,  having  a diameter  of  as  much  as  19  or  24 
inches,  but  it  was  found  very  difficult  to  get  a sharp 
image  when  they  were  used,  and  the  negative  was  often 
broken  by  the  intense  heat  which  resulted. 

To  obtain  the  greatest  illumination  possible,  all  the 
light  from  the  condenser  should  pass  through  the  nega- 
tive and  fall  on  the  objective,  and  the  position  of  the 
negative  must  be  that  shown  in  Fig.  94. 

If  we  have  given  the  number  of  times  the  picture  is 
enlarged,  the  size  of  the  negative,  the  focal  length  of 
the  objective,  and  the  diameter  of  the  condenser,  we  can 
calculate  the  focal  length  of  the  condenser  required, 
supposing  that  the  condenser  concentrates  light  to  the 
nodal  point  of  incidence  of  the  objective  (§  152). 

143.  Modern  Arrangements. — Now  that  both  plates 
and  paper  have  been  made  extremely  sensitive,  it  is  no 
longer  important  to  secure  a very  intense  illumination, 
and  a large  condenser  is  no  longer  necessary.  The 
negative  is  now  placed  close  to  the  condenser,  and,  in 
consequence,  the  diameter  of  the  condenser  needs  to  be 

* only  slightly  greater  than  the  length  of  the  diagonal  of 
the  negative ; this  very  much  reduces  the  size  of  the 


300 


PHOTOGRAPHIC  OPTICS 


apparatus.  The  remarks  made  above  about  the  form- 
ation of  the  image  will  hold  good  here  also. 

The  form  of  the  condensing  lens  now  usually  employed 


is  shown  in  Fig.  95  ; it  consists  of  two  plane  convex 
lenses  placed  with  their  convexities  turned  towards  each 
other.  The  following  are  the  dimensions  of  a condens- 


ENLARGEMENT,  REDUCTION,  ETC. 


301 


ing  lens  used  with  an  optical  lantern  to  take  slides  of 
the  standard  size,  3^  inches  square  : 

Radii  of  curved  surfaces  = 3 inches,  very  nearly. 

' Thickness  of  lenses  at  centre  = *8  inch. 

Diameter  of  lenses  = 4 inches. 

Distance  between  plane  faces  = 2 inches. 

Approximate  focal  length  of  the  combination  = 3 
inches. 

The  spherical  aberration  was  found  to  be  consider- 
able. 

Lenses  of  larger  diameter  have  their  dimensions  pro- 
portional to  those  above. 

The  lenses  should  fit  loosely  in  their  mounting,  other- 
wise they  may  be  broken  by  the  heat  from  the  source  of 
light. 

144.  Illuminatior.  without  a Condenser. — A condenser 
may  be  dispensed  with  when  intense  illumination  is  not 
required,  provided  the  negative  can  be  uniformly  illu- 
minated by  other  means.  The  required  illumination  can 
in  some  cases  be  obtained  by  placing  a sheet  of  ground 
glass  between  the  negative  and  the  source  of  light;  the 
light  coming  from  an  opal  or  uniformly  frosted  globe  of 
a lamp  might  in  some  cases  be  suitable.  It  is  impos- 
sible to  say  exactly  when  such  an  arrangement  will  be 
suitable  and  when  not — that  can  be  decided  only  by 
experiment ; but  those  who  cannot  afford  expensive 
apparatus  will  find  something  of  the  kind  well  worthy 
of  a trial. 

When  using  daylight  a condenser  can  easily  be 
dispensed  with,  if  outside  the  window  is  fixed  a board 
covered  uniformly  with  white  paper  or  cloth,  having  one 
side  horizontal  with  the  plane  of  the  board  inclined  at 
an  angle  of  45°  to  the  vertical ; the  white  surface  will 
then  be  illuminated  by  the  sky,  and  will  act  as  a uniform 
luminous  background  for  the  negative.  The  board  must 
of  course  be  large  enough  to  illuminate  the  whole 
picture,  or  unequal  printing  in  the  enlargement  will 
result. 


302 


PHOTOGEAPHIC  OPTICS 


145.  Daylight  Enlarging  Apparatus. — As  it  may 
prove  of  interest,  a description  will  be  given  of  the 
method  of  enlarging  by  daylight,  using  the  camera  in 
which  the  negative  was  taken,  when  a room  whose 


window  can  be  blocked  with  a shutter  is  available.  A 
section  of  the  arrangement  is  shown  in  Fig.  96. 

Outside  the  window  is  placed  the  board  covered  with 
white  paper  to  act  as  the  source  of  light ; just  inside 
the  window  (the  framework  of  which  is  not  shown)  is 
fixed  the  shutter  with  a shelf  to  support  the  camera ; 
the  ground  glass  is  replaced  by  the  negative  to  be 


ENLARGEMENT,  REDUCTION,  ETC. 


303 


enlarged,  and  the  end  of  the  camera  passes  through  a 
hole  in  the  shutter ; a cloth  carefully  placed  round  the 
camera  will  make  the  arrangement  light-tight.  Below 
the  camera  is  a bench  on  which  can  slide  the  screen  to 
carry  the  sensitive  paper. 

Another  hole  must  be  made  in  the  shutter,  and 
covered  with  a medium  that  admits  safe  light  only  so 
that  there  may  be  light  enough  to  make  the  necessary 
adjustments  ; the  development  can  then  be  conveniently 
carried  out  in  the  enlarging  room. 

The  routine  will  be,  the  ground  glass  is  replaced  by 
the  negative,  the  camera  is  placed  in  position,  and  the 
image  formed  is  thrown  on  the  screen,]  which  is  at 
present  covered  with  white  paper  only  ; the  position  of 
the  screen  and  the  extension  of  the  camera  necessary 
to  obtain  a picture  of  the  size  required  are  found  by 
trial — this  can  be  done  without  much  trouble.  The  cap 
is  now  placed  on  the  lens  and  the  position  of  the  screen 
is  noted  (as  explained  in  | 148),  so  that  the  screen  can 
be  removed  and  replaced  in  its  original  position  without 
trouble  ; the  cap  is  then  placed  on  the  lens,  the  sensitive 
paper  placed  on  the  screen,  the  exposure  is  made,  and 
the  picture  developed. 

Some  photographers  have  focussed  the  picture  on  the 
sensitive  paper  itself  by  using  a cap  to  the  lens  in  the 
centre  of  which  is  inserted  a piece  of  yellow  or  canary 
glass,  so  that  the  light  thrown  on  the  paper  is  not 
photographically  active ; but  there  is  no  need  for  this 
if  the  screen  can  easily  be  replaced  exactly,  when 
removed. 

146.  Geometrical  Constraints. — It  is  of  importance 
that  the  movable  screen  which  carries  the  sensitive 
paper  should  be  easy  of  adjustment,  so  that  it  may  be 
readily  placed  in  position  with  its  plane  perpendicular 
to  the  axis  of  the  lens ; also  it  should  be  possible  to 
remove  it,  to  fasten  the  sensitive  paper  to  it,  and  to 
replace  it,  in  the  dark,  in  the  exact  position  it  originally 
occupied. 


304 


PHOTOGRAPHIC  OPTICS 


Many  pieces  of  apparatus  sold  leave  much  to  be 
desired  in  these  respects,  the  screen  being  fixed  to  a 
base  which  slides  in  grooves  and  wobbles  unless  the 
fitting  is  very  good  ; some  even  run  on  wheels,  which 
offer  every  opportunity  for  shaking. 

All  unsteadiness  may  be  avoided,  ease  of  adjustment 
attained,  and  accurate  fitting  dispensed  with  by  attention 
to  the  geometrical  principles  of  the  case  ; these  principles 
are  well  known  and  are  applied  to  the  construction  of 
scientific  apparatus.^  It  can  be  shown  by  geometry 
that  a rigid  body  perfectly  free  is  capable  of  six  distinct 
and  independent  movements,  and  that  any  movement 
whatever  can  be  effected  by  a combination  of  these 
elementary  movements.  The  six  movements  are,  three 
displacements  parallel  to  three  fixed  directions  at  right 
angles,  and  three  twists  about  axes  at  right  angles.  A 
perfectly  free  rigid  body  is  thus  said  to  have  six  degrees 
of  freedom  ; any  constraint  that  is  applied  will  reduce 
the  number  of  possible  elementary  motions  or  degrees 
of  freedom. 

For  instance,  if  one  point  of  the  body  be  fixed  it 
loses  three  degrees  of  freedom,  for  the  only  elementary 
motions  left  are  the  three  twists.  If  one  more  point  is 
fixed  two  more  degrees  of  freedom  are  destroyed,  for  the 
body  can  now  rotate  only  about  the  line  joining  the  two 
points  ; if  one  more  point,  not  in  the  same  straight  line 
with  the  other  two,  is  fixed,  the  body  will  be  completely 
fixed. 

It  is  not,  however,  easy  to  fix  a point  of  a body,  it  is 
more  convenient  to  make  various  points  of  the  body 
bear  against  a surface  or  surfaces.  When  a rigid  body 
touches  a smooth  surface  at  one  point  one  degree  of 
freedom  is  lost ; for  instance,  a sphere  touching  a 
smooth  plane  cannot  move  in  a direction  perpendicular 
to  the  plane,  but  it  can  twist  about  any  three  directions 
at  right  angles,  and  it  can  be  displaced  in  any  two 

1 Thomson  and  Tait’s  Natural  Philosophy.  Ed.  1879.  Vol.  i. 
pp.  150 — 155. 


ENLARGEMENT,  REDUCTION,  ETC. 


305 


directions  at  right  angles  parallel  to  the  plane.  We 
conclude  then,  that  to  completely  fix  a body  it  must 
bear  against  other  bodies  which  are  fixed,  at  six  points  ; 
and  that  for  each  bearing  point  less  there  remains  one 
degree  of  freedom. 

Take  for  example  a three-legged  stool  in  the  middle 
of  a level  floor,  one  point  of  each  leg  is  in  contact  with 
the  ground ; it  can  be  displaced  in  any  two  horizontal 
directions  at  right  angles  and  can  be  twisted  about  a 
vertical  axis,  without  being  raised  off  the  ground.  Now 
let  the  stool  be  pushed  against  one  of  the  side  walls  of 
the  room,  so  that  two  of  its  legs  are  in  contact  with  the 
side  wall,  then  there  are  five  bearing  points  and  the 
stool  can  only  be  slid  parallel  to  the  wall  if  the  bearing 
points  keep  all  in  contact.  If  now  the  stool  be  slid 
along  till  one  of  the  legs  touches  a second  wall,  then 
there  are  six  bearing  points  and  the  position  of  the 
stool  is  completely  determined.  If  the  stool  be  removed 
from  this  position  it  can  be  exactly  replaced  by  bringing 
all  the  six  bearing  points  in  contact  with  the  walls  and 
the  floor,  an  operation  which  can  be  performed  equally 
well  in  light  or  in  darkness. 

We  see  then  that  six  bearing  points  only  are  required 
to  completely  determine  the  position  of  a body ; if  there 
are  to  be  more  points  of  contact  their  positions  cannot 
be  independent,  and  if  there  is  not  to  be  shaking  or 
wobbling,  the  fitting  must  be  good.  For  instance,  with 
a four-legged  stool,  since  three  points  of  contact  are 
enough  to  rest  a body  steadily  on  a plane,  all  four  legs 
will  not  be  in  contact  with  the  floor  unless  they  are 
made  of  the  proper  length.  In  other  words,  any  three- 
legged  stool  will  rest  steadily  in  contact  with  a plane 
without  wobbling  (provided  of  course  that  the  line  of 
action  of  the  resultant  weight  falls  within  the  triangle 
formed  by  the  points  of  contact),  but  if  there  are  more 
than  three  legs,  careful  fitting  is  required  to  make  the 
stool  rest  steadily  on  all  the  legs. 

But  we  want  not  only  to  be  able  to  fix  a body 

X 


306 


PHOTOGRAPHIC  OPTICS 


definitely,  but  also  to  effect  the  necessary  adjustments ; 
this  can  be  done  by  varying  the  positions  of  the  six 
bearing  points  relative  to  the  body  by  means  of  screws. 


ENLARGEMENT,  REDUCTION,  ETC. 


307 


147.  Application  to  Movable  Screen. — To  obtain  the 
required  movable  screen  we  must  adopt  the  principle 
of  the  three-legged  stool  with  the  modifications  necessary 
to  suit  it  to  our  case.  The  arrangement  is  shown  in 
Fig.  97 ; the  base  board  carrying  the  screen  is  sup- 
ported on  three  screws,  A,  B,  C,  with  round  ends, 
which  rest  on  the  table.  Along  one  side  of  the  table  is 
fixed  a straight  board  parallel  to  the  required  direction 
of  motion ; to  the  base  board  are  fixed  two  horizontal 
screws  D,  E which  bear  against  the  raised  edge. 

If  all  five  screws  are  kept  in  contact  the  screen  will 
have  only  one  degree  of  freedom,  i.  e.  it  can  slide 
parallel  to  the  straight  raised  edge.  To  completely 
determine  the  position  of  the  screen  one  more  bearing 
point  is  wanted ; to  supply  this  there  is  a sixth  screw 
F,  which  works  in  a block  of  wood  which  can  be  clamped 
to  the  side  of  the  straight  board  as  shown,  the  screw 
being  parallel  to  the  direction  of  sliding.  The  screen 
can  now  be  moved  till  the  base  comes  into  contact  with 
the  screw  F,  and  there  being  now  six  points  in  contact 
the  position  is  completely  determinate.  If  the  screen 
be  removed  it  can  be  replaced,  even  in  the  dark, 
by  bringing  all  the  six  bearing  points  again  into 
contact. 

If  desired,  the  screws,  as  shown,  can  be  dispensed 
with,  and  metal  screws  with  round  heads  used,  placed 
so  that  the  round  heads  form  the  bearing  points. 

148.  The  Adjustments  of  the  Movable  Screen. — The 
screws  are  used  to  adjust  the  position  of  the  screen  ; 
there  are  three  of  them.  A,  B,  C,  resting  on  the  hori- 
zontal plane  ; if  A be  turned  the  screen  is  turned  about 
a horizontal  axis  perpendicular  to  the  plane  of  the 
sensitive  paper ; if  B or  C be  turned  the  screen  is 
inclined  forwards  or  backwards,  and  by  means  of  D and 
E it  can  be  twisted  about  a vertical  axis.  Lastly, 
when  the  position  for  sharp  focus  is  found  the  sixth 
screw  can  be  clamped  so  that  it  just  bears  against  the 
base,  and  a small  final  adjustment  can  be  made  -by 


308 


PHOTOGRAPHIC  OPTICS 


turning  the  screw.  Thus  every  possible  adjustment 
can  be  easily  made. 

We  have  now  shown  how  geometrical  principles 
properly  applied  prove  of  great  use  in  designing  a 
proper  sliding  screen  ; the  same  principles  can  be  applied 
to  many  kinds  of  apparatus,  with  the  result  in  most 
cases  of  greatly  cheapening  and  steadying  them. 

149.  Enlarging  with  a Box. — When  a room  is  not 
available  daylight  enlarging  can  be  conducted,  though 
not  so  conveniently,  by  means  of  a box ; the  design  is 
shown  in  Fig.  98,  from  which  it  will  be  seen  that  the 
ordinary  camera  is  again  used. 

The  general  arrangement  does  not  need  much  explan- 
ation. The  ground  glass  is  replaced  by  the  negative  and 
the  sensitive  paper  is  placed  on  the  panel  indicated  by 
the  dotted  lines  ; the  box  is  then  held  up  to  a window 
or  lamp  so  that  the  light  falls  on  the  negative,  and  the 
sliding  shutter  inside  the  box  is  withdrawn  for  the 
time  of  exposure.  The  camera  is  fixed  to  a ledge 
which  can  be  folded  up  against  the  box  when  not  in 
use  ; the  panel  carrying  the  sensitive  paper  is  held  in 
grooves  in  the  sides  of  the  box,  its  distance  from  the 
lens  depending  on  the  size  to  which  it  is  required  to 
enlarge. 

The  box  is  rendered  light-tight,  not  by  a close-fitting 
lid  which  might  warp,  but  by  overlapping  edges  and  an 
inside  board  resting  on  a ledge  which  extends  all  round 
the  box ; there  are  then  five  corners  which  must  be 
turned  by  any  light  which  penetrates  to  the  interior. 
The  whole  of  the  inside  of  the  box,  the  lid,  and  the 
inner  board  are  painted  a dead  black,  to  prevent  the 
reflection  of  stray  light. 

Light  is  prevented  from  entering  at  the  lens  hole, 
round  the  sides  of  the  lens  mounting,  by  a flange  which 
just  overlaps  the  mounting  and  is  painted  dead  black. 
Some  difficulty  may  be  found  in  focussing  the  picture 
properly,  but  when  once  it  is  found  the  trouble  may 
be  avoided  by  marking  on  the  tail-board  of  the  camera 


ENLARGEMENT,  REDUCTION,  ETC. 


309 


the  positions  of  the  negative  corresponding  to  various 
positions  of  the  panel. 


310 


PHOTOGRAPHIC  OPTICS 


The  enlarging  box  described  has  the  convenience  that 
the  whole  outfit  can  be  packed  inside  it  for  travelling. 

150.  Reducing  Apparatus. — It  is  often  required  to 
reduce  a picture  to  form  a lantern  slide ; the  optical 
principles  of  reduction  are  exactly  the  same  as  those  of 
enlarging,  and  the  relative  distances  can  be  calculated 
in  a similar  manner.  The  enlarging  box  described  in 
the  last  article  may  be  used  for  reducing,  provided  the 
camera  can  be  extended  enough  and  a board  carrying 
the  lantern  plate  can  be  placed  at  a suitable  distance. 

When  designing  apparatus  for  enlarging  or  reducing, 
care  should  be  taken  to  use  the  correct  focal  length  of 
the  lens,  for  an  error  of  even  half-an-inch  will  make  a 
great  deal  of  difference  in  the  relative  positions  of 
negative  and  enlargement.  The  focal  lengths  given  by 
makers  in  their  catalogues  are  often  the  back  focus,  and 
not  the  true  focal  length. 

151.  Distances  of  Negative  and  Enlargement  from 
the  Lens. — The  distances  of  the  negative  and  the 
enlargement  from  the  nodal  points  of  the  lens  can  be 
calculated  from  the  principles  explained  in  Chapter  II. 
If  u and  V be  the  distances  of  the  negative  and  en- 
largement, y the  focal  length  of  the  lens,  and  n the 
linear  magnification  required,  we  have 


1 

V 


V 

u 


n (2) 


the  negative  sign  being  used  in  (2)  as  the  picture  is 
inverted. 

Solving  for  v and  u we  get 

u = — (1  + 7^)  fjn^  'y  = (1  + n)f 

Example, — With  a lens  of  8 inches  focal  length  it  is 
required  to  enlarge  from  a quarter  plate  negative  to 
five  times  the  linear  dimensions. 


Here  / = — 8 inches,  n = 5 

,\u  = + 5)  X 8/5  = 48/5  = 9’6  inches, 

and  = (1  + 5)  X 8 = — 48  inches; 


ENLARGEMENT,  REDUCTION,  ETC. 


311 


or  the  negative  must  be  placed  at  a distance  of  9 ‘6 
inches  from  the  nodal  point  of  incidence,  the  paper  to 
receive  the  enlarged  picture  at  a distance  of  48  inches 
from  the  nodal  point  of  emergence.  The  table  on  next 
page  gives  the  relative  distances  for  lenses  of  various 
focal  lengths ; in  the  vertical  column  on  the  left  is 
given  the  principal  focal  length  of  the  lens,  and  along 
the  top  the  linear  ratio  of  enlargement.  The  distances 
required  are  read  off  in  the  usual  manner. 

The  results  will  clearly  hold  good  in  whatever  units 
the  lengths  are  measured  as  long  as  all  three  are  at  one 
time  in  terms  of  the  same  unit. 

This  table  can  also  be  used  for  reduction,  for  we  have 
only  to  interchange  the  places  of  the  negative  and 
sensitive  surface ; for  instance,  in  the  example  above,  if 
the  negative  be  placed  at  a distance  of  340  inches  and 
the  sensitive  surface  at  a distance  of  17*9  inches,  the  pic- 
ture will  be  reduced  to  one-nineteenth  of  its  original  size. 

Example, — The  focal  length  of  the  lens  is  1 7 inches 
and  it  is  required  to  enlarge  19  times. 

At  the  row  containing  1 7 and  the  column  containing 
19  ^ we  find  the  numbers  340,  17*9,  which  mean  that 
the  negative  must  be  distant  17*9  inches  and  the  en- 
largement 340  inches  from  their  respective  nodal  points. 

152.  Times  of  Exposure  when  Enlarging  or  Reducing. 
— We  have  seen  in  § 1 1 9 that  if  T be  the  time  of  exposure, 
u the  distance  of  the  object,  and  e that  of  the  stop  from 
the  nodal  point  of  incidence,  d the  diameter  of  the 
aperture  in  the  stop,  then 


If  e is  small  enough  to  hp  npo-lpcted  then 

2 


Let  Ti  be  the  exposure  with  distance  and  diameter 
di,  then  we  get 


Tj  “ ^ 


TABLE  FOR  ENLARGEMENT  AND  REDUCTION. 


iO 

G<l 

2i> 

2" 

208 

8*3 

2 

CO  • 

CM  ^ 

260 

10-4 

1286 

11-4 

312 

12-5 

> 

1125 

5-2 

rH  CO 

CM  «> 

lO  ^ 
CM  J 
(M 

250 

10-4 

275 

11-5 

300 

12-5 

H 

Q 

W 

CO 

oi 

120 

5-2 

CO 

1— 1 ^ 

22 

2^ 

2 ^ 
CM 

O rH 

§!s 

264 

11  5 

288 

12*5 

o 

w 

Cv| 

cq 

o 

rH 

^ S 

rH  'O 

2 2 

184 

8-4 

Si 

(M 

230 

10-5 

253 

11*5 

27 

12-5 

w 

21 1], 

no 

i 5*2 

CO 

22 

154 

7-8 

rH  CO 

2o 

O kO 

§:!s 

242 

11-5 

2641 

12*6 

o 

m 

H 

20  t 

^ CO 
2*^ 

§s 

S7 

2 00 

189 

1 9*5 

210 

10-5 

2311 

11-6 

(M  CO 

JD  r<j 

CM  H 

g 

O 

Ph 

Gi 

ss 

r— 1 ^ 

Si 

§7 

8?- 
Ph  oo 

Sg 
1— ! ^ 

200 

10-5 

220* 

11-6 

240 

12-6 

p 

00 

JO 

Ci  lio 

rH 

2 ^ 

g? 

rH  00 

f-H  ^ 

190 

10*6 

209 

11-6 

GO  J- 

8kS 

b 

Iz; 

w 

H 

D- 

O 

05  o 

Sr 

pH 

CD  ^ 
^2 

144 

8*5 

162 

9-5 

180 

10-6 

198 

11-6 

216 

12-7  ; 

CO 

rH 

JO 

00  o 

102 

6*4 

611 

2i) 

153 

9-6 

170 

10-6 

187 

11-7 

Tt(  Jb- 

O 

Ph 

lO 

O 

GO  o 

OJ 

G5  o 

128 

8-5 

9-6 

ffl 

O 1- 

Ss 

176 

11-7 

192 

12-8 

W 

lO  T*^ 

lO 

O 

05  o 

22 

2 <» 

135 

9-6 

150 

10-7 

165 

11-8 

180 

12-9 

CO 

O 

J>»  o 

GO  o 

GO  p 
05 

2 ^ 
2 ^ 

§s 

140 

10-8 

154 

11-8 

168 

12-9 

> 

H 

CM 

JO 

CD  o 

GO  p 

CO 

rH  p 
G5  lb 

sS 

rH 

130 

10*8 

143 

111-9 

156 

13 

OQ 

;zi 

w 

m 

W 

O 

CD  o 

C4  p 

!>»  CO 

p 
GO  ^ 

CO  ^ 
05  00 

108 

9-8 

120 

10-9 

CO  ^ 

144 

13-1 

o 

JO  ^ 
JO  o 

CO 

CD  cb 

!>• 

1>*  Jb- 

GO  00 
00  i) 

C5  p 
05  C5 

o ^ 

121 

12-1 

132 

13-2 

Q 

Iz; 

o 

O 'p 
JO  kO 

O ^ 
CD  CO 

O CO 

o 00 
GO  (b 

o p 

05  C5 

100 

11-1 

no 

12-2 

120 

13-3 

<J 

p> 

NO 

00 

JO 

rf  P 
JO  cb 

CO  Ci 
CD  t- 

72 

9 

81 

10*1 

90 

11-3 

99 

12-4 

108 

13*5 

H 

b 

N<J 

!>- 

O 1- 

o 

GO  Oi 
^ cb 

56 

8 

rjH  -H 
CO  05 

72 

10-3 

80 

11-4 

88 

12-6 

96 

13-7 

w 

iz; 

w 

w 

H 

CO 

JO  p 
CO  ko 

42 

7 

05 

00 

CO  00 
JO  b 

63 

10*5 

70 

11-7 

77 

12-8 

r}^ 

GO  rH 

NO 

lo 

30 

6 

CO 

CO  J^ 

CM 

NH  cx) 

GO  P 

Ttl  05 

54 

10-8 

O (M 
CO  rH 

66 

13-2 

72 

14-4 

P^ 

O 

CO 

W 

Q 

!z; 

JO 

(M  cb 

o ^ 
CO  L- 

JO  p 
CO  oo 

o o 

^ rH 

45 

11-3 

o ‘P 

JO  c<J 

55 

13-8 

60 

15 

CO 

0 

01  CO 

24 

8 

GO  CO 
CM  o 

32 

10-7 

CO  <N 
CO  rH 

40 

13-3 

44 

14-7 

GO  CO 

TH  rH 

5 

NO 

Ol 

JO  P 
rH  t- 

18 

9 

rH  P 

CM  o 

C<J 

CM  rH 

t-P 
(M  ^ 

O kO 

CO  rH 

33 

16*5 

CD  00 

CO  rH 

rH 

O O 

04  CO 

tH 

CO  CO 

GO  OD 

O O 

CM 

(M  (M 
(M 

CM 

•S1191  911'!^  JO  1[J§119T^ 
X«00^  J-BtllOUlIJ 

JO 

CO 

GO 

05 

O 

- 

CM 

QO  lO 

0 

CD  CO 

Q1  t- 

00  b- 

. rjfH  (X 

i 0 <50 

(^  <0 

CM  rH 

GO  C5 

0 

t- 

2 05 

C5 

, d i 

,5 

b 

R?  b 

.r, 

RJ  CM 

XQ  g<7 

CO  S 

CO  ^ 

CO  r-5 

5h 

5; 

r^ 

rH 

XQ  Gs, 

X Q CM 

XQ  CM 

XQ  07 

CD 

CD 

0 1- 

XQ 

0 CO 

XQ  CO 

0 CO 

XQ  05 

0 <35 

^ rH 

^ .0 

*0  4^ 

iri 

j5 

^47  b 

s 0 

01  4^ 

^ b 

CO 

CO  S 

CO 

Tt^  S 

TfH  rH 

rH 

rH 

XQ  ^ 

XQ  07 

XQ  07 

XQ 

0 

ZO 

(M  «o 

CD  0 

0 

00  b- 

d CO 

CD  00 

0 05 

05 

GO  ^ 

C'l  ^ 

^ XO 

0 rH 

CO  41^ 

^ ‘b> 

00  cb 

2^ 

£2  i5 

*2  b 

22  b 

0 4h 

CM 

^R  07 

SSb 

CO  ” 

CO 

CO  r4 

CO  r5 

r- 

tJH  rH 

rH 

0 CM 

XQ 

XQ 

CD  07 

C5  O 

D1  0 

10  Jt- 

00  X- 

r-H  00 

r^t  CO 

1>*  CO 

0 05 

CO  <35 

^ A-. 

CM  rH 

XQ  ph 

cb 

(M 

0 

SS 

CJ5 

22  b 

SR  b 

00  4h 

CO 
^ C<l 

-H 

Cl  ^ 

!R  b 

f b 

j— ( 

CO  r4 

CO  r4 

CO  rH 

CO  rH 

rH 

07 

10 

XQ 

XQ  CM 

XQ  CM 

zo  -X) 

00  i- 

0 t- 

Cl  05 

CO 

CD  CX5 

00  05 

0 _ 

0^1 

Hf( 

CD  rH 

GO  rH 

0 CM 

0 

»Q  i, 

4, 

rH  p. 

07 

GO  ?2 

0 4h 

d b 

^ b 

(M  S 

CO 

CO  rH 

CO  r5 

CO  rH 

CO  5; 

5h 

r^ 

XQ  b 

XQ  CM 

XQ  C7 

CO  t- 

Tin  i- 

10  00 

CD  00 

l>*  a 

GO  crq 

0 

I-H  rH 

CM  rH 

CO  IM 

r^^  CM 

XQ  CO 

r— 1 ^ 

CO  i, 

XQ  ,L 

R]  Sq 

SR  b 

GO  4h 

^ b 

CM  b 

<M  r4 

01  3 

CO  15 

CO  5; 

CO  rH 

CO  r4 

CO  ^ 

CM 

CM 

Tf  CM 

XQ  CM 

XQ  b 

O 

0 t- 

0 GO 

0 05 

0 0 

0 cn 

0 ^ 

0 rH 

0 rH 

0 CM 

0 07 

0 CO 

0 CO 

Q1  0 

cb 

0 4h 

Rj  b 

b 

4h 

22  b 

S b 

(OJ  r4 

(01 

CO  r4 

CO  5- 

CO  rH 

CO  rH 

CO 

<^7 

CM 

TTI  07 

CM 

XQ  07 

CD  00 

IQ  CO 

Tfl  Ci 

(CO  Oi 

i-H  rH 

0 <b7 

(35  <>7 

GO  CM 

!>•  CO 

CD  CO 

XQ  rH 

CO 

CO  4ii 

GO  0 

0 0 

r-, 

5f(  2 

^ b 

00  4^ 

D5  b 

I-H  4-, 

CO  b 

^ vO 

b 

(M  S 

(M 

Cl  r5 

CO  r5 

CO  5h 

CO 

CO 

CO  CM 

CO  (M 

^ 07 

b 

b 

Tf^  00 

(M  05 

0 0 

00  01 

Htn  rH 

Q1  rH 

0 CM 

00  07 

CD  07 

<p 

CM  rH 

0 vO 

CO  CO 

iR 

0 92 

SSI  0 

r5 

07 

b 

GO  b 

‘R  b 

(M  rH 

Cl 

Cl  rH 

(Q1  rH 

CO 

CO  rH 

CO  CM 

(CO  07 

CO  CM 

CO  07 

b 

i—l  GO 

00  Oi 

XQ  0 

Cl  o> 

C5  rH 

CD  i-l 

CO  (M 

0 CO 

!>•  CO 

rH 

i-H  tH 

GO  vo 

XQ  0 

JR  0 

0 

,5 

JR  b 

b 

C5  4fH 

O'  b 

d b 

(M  r4 

Cl 

Cl  rH 

Cl  rH 

(M  rH 

CO  rH 

CO 

CO  CM 

CO  ^ 

CO 

CO  b 

tJh  b 

TtH  b 

00  05 

Tt<  CSi 

CD  rH 

Cl  rH 

00  CM 

rrfH  CO 

0 CO 

0 rH 

CM  VO 

GO  vo 

r^^  'O 

0 Xh 

s * 

)R^ 

^ 00 

s 0 

Cl  4h 

22  b 

JR  b 

GO  b 

0 b 

G<1  rH 

Cl  rH 

Cl 

Cl  rH 

(Q1  rH 

d rH 

CO  (M 

CO  07 

CO  CM 

CO  CM 

CO  b 

CO  CM 

^ b 

lO  C5 

0 

rH 

0 rH 

XQ  07 

0 CO 

XQ  rH 

0 rH 

XQ  >0 

0 0 

XQ  CO 

0 X- 

XQ  CO 

JR  w 

22  b 

0 4h 

I-H  b 

GO  b 

^ b 

Cl 

Cl  rH 

Cl  rH 

d rH 

d rH 

Ql  CM 

CO  CM 

CO  M 

CO  07 

CO  b 

CO  b 

CO  b 

CD  rH 

0 oq 

(M 

00  CO 

(M  rH 

CD  0 

0 >C0 

rcfH  *0 

GO  JH 

CM  <50 

CD  CO 

0 <35 

00 

05  4^ 

]r? 

^ (So 

JR 

SR  0 

GO  4h 

05 

S b 

CM  b 

GO  b 

XQ  b 

r-H  rH 

Q1  rH 

Cl  rH 

d r-, 

d rH 

d CM 

CM  CM 

(Q1  ox 

CO  ^ 

CO  b 

CO  b 

CO  b 

05  rH 

Cl  oq 

xo  cp 

00  CO 

I-H  Ttl 

T)H  lO 

J>“  «o 

0 b- 

CO  00 

CD  CO 

<35  05 

XQ  rH 

2 

2 

05 

RJ  ^ 

0 

,5 

b 

GO  b 

C5  4J^ 

CM 

I-H  rH 

Cl  rH 

d rH 

d rH 

(d  <M 

d 07 

d b 

d CM 

CM  CM 

CO 

CO  CM 

CD  (M 

GO  CO 

0 

Cl  0 

0 

CD  p 

GO  X- 

0 CO 

d 05 

CD  rH 

GO  CM 

0 CO 

»o 

CD  0 

00  i. 

2^ 

2 ^ 

'zi  05 

0 

^ r5 

JR  b 

SR'  ^ 

GO  b 

0 jC, 

j— t 

I-H  rH 

d rH 

Ql  rH 

d CM 

(d  CM 

d 07 

d 

d b 

d b 

CO  CM 

CO  CO 

XQ  0 

CD  0 

jt- 

00  GO 

<35  05 

0 rv, 

rH  rH 

d CM 

CO  CO 

Tfl  rH 

XQ  vo 

CD  0 

2 ^ 

2 s 

S ^ 

22  b 

^ b 

JR  b 

SR  b 

ir:  ^ 

1— 1 r-l 

r~i 

r— t 

r— 1 rH 

d 07 

d 

CM  CM 

(d  CM 

CM  CM 

d CM 

d CM 

O tJ< 

0 0 

0 x^- 

0 00 

0 0 

0 _ 

0 rH 

0 CM 

0 <p 

0 rH 

0 0 

0 ]H 

0 <50 

CO  ;,^ 

21 

XQ  4^ 

2^ 

2"  s 

2 ^ 

C5  4^ 

JTI  CO 

22  b 

^ b 

JR  ^ 

r—i  rH 

rH 

1 — 1 CM 

d CM 

d CM 

d CM 

(Q1  07 

CM  CM 

d CM 

!>•  O 

(:d  00 

xQi  0 

CO  ^ 

Q1  00 

p-H  rH 

0 iO 

<35  w 

00  CO 

!>•  05 

CD 

XQ  rH 

CO  ^ 

2 s 

2o 

,5 

GO  b 

GO  cb 

4h 

S b 

2^  b 

<— 1 rH 

>— ( CM 

i-H  (OJ 

rH  07 

I-H  07 

d (M 

d 

CM  07 

Oi 

0 rH 

00  CO 

CD  rH 

d JH 

0 C35 

GO  ^ 

CD  rH 

CO 

Q1  rH 

0 0 

O ^ 

r-H  'C 

Cl 

2 2 

2s 

^ b 

XQ  4^ 

^ b 

2 ^ 

VO 

GO  b 

05  4, 

0 b 

>— 1 rH 

r-H  CM 

I-H  CM 

1— ( CM 

I-H  CM 

1— ( 07 

1— 1 CM 

d CM 

rr^  CO 

XQ  lO 

Cl  ^ 

05  CO 

CD  ^ 

CO  CM 

0 CO 

!>•  VO 

r^l  b- 

r-H  i>. 

GO  ^ 

XQ  CM 

00  • 

05 

0 ^ 

1 ( 00 

2 ^ 

2 

CO  (Cq 

^ cb 

^ 4h 

b 

CD  b 

2 ^ 

^ b 

1— 1 rH 

i-H  CM 

I-H  07 

c-H  CM 

I-H  07 

I-H  CM 

I-H  07 

00-^ 
J^  o 

tH  90 

00  'C 

90 

18 

96 

9-2 

d rH 

0 0 

GO  0 

0 4^ 

-ich  Cp 

' ' CM 

HH 

d c<) 

i^l  CM 

d b 

d 

CO  b 

GO  CO 
CO  jb 

GO 

^ (b 

i” 

rH  CM 

>— ( CM 

I-H  CM 

1 — 1 CM 

I-H  07 

1 — 1 CM 

lO 

0 ‘P 

XQ  ^ 

0 0 

XQ 

0 ‘P 

XQ  <» 

S 0 

XQ  CO 

0 vO 

XQ  <50 

XQ  (n 

1:0  'C 

!>.  05 

00 

GO  Ti 

05  2^ 

<35  fO 

2 

0 b 

b 

CO 

d 4h 

CM 

CM 

CM 

1 — 1 CM 

1 — 1 CM 

I-H  07 

I-H  CO 

CD  ^•'~ 

0 0 

9^ 

00  ^ 

Q1  rH 

CD  ‘P 

oP 

00 

00  P 

CM 

CD  CM 

0 CO 

10  ^ 

10  2 

CD  (M 

ZO  rH 
^ CM 

!>.  c>q 

t-r.  lO  . 
^ CM 

00  <^^ 

<35  cXi 

S b 

1 — ( CO 

05  ‘P 

Cl  rH 

XQ'P 

00  riH 

p-H  ‘G’ 

jr-P* 

0 0 

CO  P 

CD  CO 

C5  P 

d CD 

XQ  P 

CO  ^ 

rji  (M 

'C'g 

XQ)  CM 

zo  CO 

^D  ^ 

ZO  CO 

ZO  rH 
CO 

!>.  CO 

J>.  IH 

CO 

SR  'C) 

00  (X5 

0 0 

Cl  07 

rH 

CD  ICO 

GO  <50 

0 0 

CM  07 

tH  -h 

CD  0 

GO  CO 

0 0 

<5^ 

d (N 

CO  CO 

CO 

CO  CO 

CO  CO 

CO  CO 

rH 

r^  rH 

-+l  rH 

Htl  rH 

rtl  rH 

XQ)  >0 

CO 

-rH 

XQ 

CO 

t-r 

CO 

<35 

0 

(M 

CO 

XQ 

r-H 

Q1 

CM 

CM 

(01 

CM 

CM 

314 


PHOTOGRAPHIC  OPTICS 


Hence  we  can  compare  the  exposures  in  two  different 
cases. 

The  quantity  u can  be  got  from  the  table  in  the  last 
section,  and  d can  be  found  from  the  focal  length  of 
the  lens  and  the  number  of  the  stop  used. 

Example. — When  using  a lens  of  7 inches  focal 
length  and  stop  y/20  to  enlarge  five  times,  the  exposure 
required  was  40  seconds ; find  the  exposure  when  a 
lens  of  9 inches  focal  length  is  used  with  the  stop  //30 
to  enlarge  six  times. 

We  find  from  the  table  that  for  the  first  case,  = 
42  inches,  and  for  the  second  case,  u = inches;  also 
in  the  first  case,  = focal  length  /20  = 7/20  inch,  in 
the  second  case  d = 9/30  = 3/10  inch,  and  the  first 
exposure  =40  seconds. 


X Ti 


X 40 


= 122  seconds,  about. 


153.  Relative  Positions  of  Condenser  and  Objective. 

— It  has  been  stated  in  § 142  that  if  the  full  advantage 
is  to  be  taken  of  the  condenser  to  concentrate  light  on 
the  negative,  certain  relations  must  exist  between  the 
focal  lengths  of  the  condenser  and  objective ; we  now 
proceed  to  investigate  these  relations.  The  results  will 
be  approximate  only,  for  it  would  complicate  the  calcu- 
lations too  much  to  take  account  of  the  spherical 
aberration  of  the  condenser  and  the  size  of  the  source, 
when  artificial  light  is  used ; we  shall  suppose  both  the 
condenser  and  objective  to  be  thin  lenses. 

In  Fig.  99,  S is  the  source  of  light,  A B the  condenser, 
C D the  diagonal  of  the  negative,  G H J K the  mounting 
of  the  objective,  E F the  lens  equivalent  to  the  objec- 
tive, here  supposed  thin,  L M N the  points  on  which  the 
common  axis  of  the  lenses  meets  the  lenses  and 
negative,  X the  point  towards  which  the  light  is 
converged  by  the  condenser,  and  Y the  point  to  which 


ENLARGEMENT,  REDUCTION,  ETC. 


315 


the  light  going  towards  X is  converged  by  the  objective, 
so  that  X and  Y are  conjugate  foci. 


316 


PHOTOGRAPHIC  OPTICS 


Let  V,  be  the  distances  of  negative  C D,  and  the 
enlargement  from  N. 

Let  U,  Y,  be  the  distances  of  S and  X from  L. 

„ 2 ^ ,,  ,,  length  G H of  the  mounting. 

,,  r ,,  5,  distance  L M. 

,,  2 X and  2 z he  the  diameters  of  A B and  E F. 

,,  2 y he  the  diagonal  of  the  negative. 

,,  F and  / be  the  principal  focal  lengths  of  the 
condenser  and  objective. 

,,  be  the  ratio  of  linear  magnification. 

Then  we  can  at  once  write  down  the  following 
relations — 


1 

Y 


U 


1 

V 


1 1 V 

U f ^ 


n 


From  the  last  two  we  find  as  in  the  last  section 


M = — (1  + n)fln,  -y  = (1  + n)f  ....  (2) 

If  the  negative  is  placed  as  in  the  figure  so  that  its 
diagonal  is  just  within  the  cone  of  rays,  we  have  the 
triangles  A L X,  C M X similar,  and  hence,  A 1 : C M = 
L X : M X 

ov  X : y = Y : Y — r (3) 

But  if  the  cone  of  light  is  to  just  fit  into  the 
mounting  as  in  the  figure  we  must  have  from  the 
similar  triangles  A L X,  G O X 

AL  : GO  = LX  : OX 


ovx'.z  — Y:Y-—v  — r + l (4) 

forOX  = OX  + XX  = ON  + LX-LM-MN  = 
I + Y -r  - V. 

These  four  relations  will  enable  us  to  calculate 
anything  we  wish. 

(a)  Take  the  case  given  in  § 142  where  sunlight  is 
used  and  it  is  required  to  find  the  focal  length  of  the 
condenser  where  y,  x,  y,  I are  given.  Here,  since  the 
sun  is  the  source  of  light,  S is  very  distant  and  we  get 


ENLAEGEMENT,  REDUCTIOlSr,  ETC. 


317 


from  (1)  F = Y.  From  (3)  and  (4)  it  can  be  shown 
that 


V — I 


2/ - ^ 
X 


but  Y = F,  and  from  (2)  = (1  + n)f 


hence 


(1  + n)f  _ y - z 


or  F = •(!+»*)  •/ 

2/  - « 


F X 

(h)  Next  consider  the  case  when  the  negative  is 
placed  close  to  the  condenser,  and  let  it  be  required  to 
find  the  distance  of  S from  the  condenser  when  z,  n, 
F,  f are  known. 

Here  r = 0 and  we  get  from  (4) 

1 


^ V — I z 1 X 

1 - - V”  = - or  - = - 

Combining  this  with  (1)  we  get 

X — z \ 1 X — z 


V - I 


1 X — 

or  - = 

V — L V X 


1^ 

F 


X V — L V X (1+  n)f  — ^ 

from  which  we  can  find  u the  distance  of  S from  the 
condenser. 

Example. — Let  the  focal  lengths  of  the  condenser 
and  objective  be  3 inches  and  6 inches  ; hence,  F = — 
3,  y = — 6 ; also  let  n = 5,  x*  = 1*5  inches,  2:  = *5  inch, 
^ = 1*5  inches,  then 


1 _ 1*5  - *5  1 1 

U “ 1-5  ^ - 36  - 1-5  3 

. *.  U = 3*17  inches. 


•3155 


Or  the  source  of  light  S must  be  3*17  inches  from 
the  condenser  to  give  the  best  illumination. 

It  should  be  noticed  that  the  nearer  S is  to  the 
condenser  the  greater  is  the  amount  of  light  which  falls 
on  the  condenser. 

154.  The  Position  of  the  Source  of  Light. — AYe  can 
use  Fig.  99  to  explain  the  faults  in  illumination  of 
the  image,  familiar  to  every  one  who  has  manipulated  a 


318 


PHOTOGRAPHIC  OPTICS 


lantern,  which  are  due  to  the  wrong  position  of  the 
source  of  light  S. 

If  S approach  the  condenser  X will  recede  from  it, 
and  some  of  the  extreme  rays  of  the  cone  will  be  cut 
off  by  the  mounting  of  the  objective.  Also  if  S recede 
from  the  condenser  X will  approach  it;  this  will  cause 
Y the  vertex  of  the  emergent  cone  to  move  towards  E E, 
and  at  the  same  time  the  angle  of  the  cone  will  widen ; 
on  both  these  accounts,  if  S be  moved  too  far  the  outer 
rays  of  the  emergent  cone  will  be  intercepted  by  the 
mounting.  In  both  these  cases  a dark  ring  will  be 
formed  round  the  edge  of  the  disc  on  the  screen. 

If  S be  moved  off  the  axis  to  one  side  it  can  easily 
be  seen  that  some  of  the  rays  will  be  intercepted  by 
the  mounting,  and  a dark  space  will  be  formed  on  the 
disc,  on  the  same  side  as  that  to  which  S was  displaced. 

155.  Depth  of  Focus. — It  has  been  pointed  out  (§  90) 
that  to  obtain  a sharp  picture  it  is  not  necessary  to 
have  the  light  proceeding  from  a point  in  the  object 
converging  exactly  to  a point  in  the  image  ; but  that  as 
long  as  the  section  of  the  refracted  pencil  by  the  plate 
does  not  exceed  a certain  definite  size  the  patch  of 
light  formed  will  appear  to  the  eye  as  a point. 

This  gives  us  some  latitude  in  focussing,  for  the 
ground  glass  can  be  moved  about,  within  certain  limits, 
without  the  section  of  the  pencil  becoming  too  large  to 
appear  as  a point ; we  shall  call  this  permissible  dis- 
placement, depth  oj-  J-ocus, 

Consider  first  the  case  when  the  object  is  at  a great 
distance  so  that  rays  converge  to  F,  the  principal  focus 
of  the  lens  (Fig.  100).  Let  F be  the  principal  focal 
length  of  the  lens,  2 e the  greatest  permissible  breadth 
of  the  patch  of  light,  and  x the  greatest  distance  from 
F to  which  the  ground  glass  can  be  moved  ; then  it  is 
clear  from  the  figure  that 


X e 

F y 


F 

e — 

y 


or  X 


ENLARGEjMENT,  reduction,  etc. 


319 


but  the  screen  can  be  moved  to  an  equal  distance  on 
the  other  side  of  F,  so  the  depth  of  focus  is 

2 X = 2 e F/i/. 


320 


PHOTOGRAPHIC  OPTICS 


Next  let  the  object  be  at  a distance  u,  and  the 
image  at  a distance  v from  the  lens  (Fig.  101)  ; then  it 
is  clear  from  the  figure  that 

X e V 

- = — or  X = e - 
V y y 

If  the  expression  is  required  in  terms  of  the  distance  of 
the  object  from  the  lens,  we  have 

uY"  Yu 

^ u + Y - ^ = ^ q.  Y. 

Hence  depth  of  focus  = 2 x = 2 e , — . — 

y u + Y 

It  should  be  noticed  that  x increases  as  y decreases, 
or  the  smaller  the  stop  the  greater  the  depth  of  focus. 

Examyile. — The  object  is  20  feet  distant,  and  the 
focal  length  of  the  lens  is  6 inches,  the  radius  of  the 
aperture  is  *5  inch,  and  2 e is  taken  as  one-fiftieth  of 
an  inch — 

1 -6  240 

Depth  of  focus  = ^ X — X — — ‘058  inch. 
^ 50  *5  246 

No  particular  meaning  can  here  be  given  to  the 
negative  sign,  as  we  have  not  said  in  which  direction  the 
depth  of  focus  is  to  be  reckoned. 

156.  Depth  of  Field.  — The  displacement  that  can  be 
given  to  a point  on  the  axis  of  the  lens,  without  the  size 
of  its  image  focussed  when  the  point  is  placed  in  a given 
position  becoming  broader  than  2 6,  is  called  the  depth 
of  field ; in  this  case  it  is  the  object  which  is  moved 
while  the  ground  glass  remains  steady. 

The  special  interest  of  the  question  lies  in  its  appli- 
cation to  hand  cameras,  where  it  is  required  to  know 
the  least  distance  of  the  object  which  will  give  a sharp 
picture. 

Let  us  consider  how  near  to  the  lens  an  object 
originally  focussed  when  at  a great  distance  can  be 
moved  without  spoiling  the  sharpness  of  the  picture. 


ENLARGEMENT,  REDUCTION,  ETC. 


In  Fig.  102,  F is  the  principal  focus  and  Q is  the 
focus  conjugate  to  P,  the  nearest  point  for  which  the 

Y 


322 


PHOTOGRAPHIC  OPTICS 


Fig,  102, 


ENLARGEMENT,  REDUCTION,  ETC. 


323 


image  is  sharp ; let  F be  the  focal  length  of  the  lens, 
let  F Q = X,  and  let  ii  be  the  distance  of  P from  IST^,  the 
nodal  point  of  emergence. 

When  the  object  is  at  P,  the  section  of  the  pencil  of 
rays  by  the  ground  glass  at  F is  2 e,  hence  we  get  from 
the  figure 

X F + .T  F -F  X y 

- = — — — or  = - 

e y X e 

But  since  P and  Q are  conjugate  foci 

_ _ 1 _ 1 . 1 _ 1 _ 1 _ .T 

F -j-  .X*  u F 1/-  F + X F F (F  + x) 

X Q 

which  gives  the  distance  of  the  object  required. 

The  following  table  gives  the  values  of  calculated 
from  this  expression  for  various  lenses  and  apertures. 

The  focal  length  is  given  in  centimetres,  the  value  of 
2 e is  taken  to  be  one  hundredth  of  a centimetre,  and 
the  results  are  given  in  metres. 


Principal 

Focal 

Aperture. 

Length  in 

Centi- 

/ 

/ 

/ 

/ 

/ 

/ 

/ 

/ 

metres. 

TTT 

TT 

■g-o- 

■Js' 

siy 

3 5 

4 5 

o’ O' 

5 5 

■G-O- 

5 

2-5 

1-3 

0-9 

0-7 

0-5 

0-5 

0-4 

0-4 

0-3 

0-3 

0-3 

0-3 

10 

10-0 

5-0 

3-4 

2-5 

2-0 

1-7 

1-5 

1-3 

1-2 

1-0 

0-9 

0-9 

15 

22-5 

11-3 

r-5 

5-7 

4-5 

3-8 

3-3 

2-9 

2-5 

2-3 

2-1 

1-9 

20 

40-0 

20-0 

13-4 

10-0 

8-0 

6-7 

5-8 

5-0 

4-5 

4-0 

3-7 

3-4 

25 

62-5 

31-3 

20-9 

15-7 

12-5 

1^ 

9-0 

7-9 

7-0 

6-3 

5-7 

5-3 

30 

90-0 

45-0 

30-0 

22-5 

18-0 

15-0 

12-9 

11-3 

10-0 

9-0 

8-2 

7-5 

35 

122-5 

61-3 

40-9 

30-7 

22-5 

20-5 

17-5 

15-4 

13-6 

12-3 

11-2 

10-3 

40 

lGO-0 

80-0 

53-4 

40-0 

32-0 

26-7 

22-9 

20-0 

17-8 

16-0 

14-6 

13-4 

45... 

202-5 

101-3 

67-5 

50-7 

40-5 

33-8 

29-0 

25-4 

22-5 

20-3 

cl8-4 

16-9 

50 

250-0 

125-0 

83-4 

62-5 

50-0 

41-7 

35-8 

31-3 

27-8 

25-0 

22-8 

20-9 

For  example,  with  a lens  of  25  cm.  focal  length  and  a 
stop  of  yy30,  all  objects  between  infinity  and  a distance  of 
10*5  metres  will  be  in  focus  at  the  same  time. 

157.  Halation  or  Irradiation. — It  sometimes  happens 
that  chemical  action  takes  place  in  parts  of  the  plate  on 
which  the  light  has  not  directly  fallen,  causing  the 


324 


PHOTOGRAPHIC  OPTICS 


phenomenon  known  to  practical  photographers  as  Hala- 
tion. Experience  shows  that  halation  depends  to  a 
great  extent  on  the  nature  and  thickness  of  the  sensitive 
surface,  and  is  specially  prominent  when  the  light  is 
very  bright,  and  the  contrasts  sharp.  If  the  image 
of  a bright  point  of  light,  such  as  the  reflection 
from  a drop  of  mercury,  be  photographed,  not  only  is 


the  ordinary  image  formed,  but  round  it,  and  separated 
from  it  by  a clear  space,  is  a dark  ring.  The  effect  of 
a bright  line  of  light  is  that  got  by  superposing  the 
effects  of  all  the  points  composing  the  line,  the  result 
being  that  the  line  is  broadened. 

The  phenomenon  has  been  explained  by  Abney,  as 
due  to  reflection  of  light  which  has  passed  through  the 
film,  at  the  back  of  the  plate;  the  reflected  light  causing 
chemical  action  at  points  around  the  regular  image. 


ENLARGEMENT,  REDUCTION,  ETC. 


325 


To  explain  this  more  clearly,  let  the  figure  (Fig.  103) 
represent  a magnified  section  of  a piece  of  the  sensi- 
tive film  and  of  the  supporting  glass  ; and  let  K be  a 
particle  of  silver  bromide  in  the  film. 

Let  A a,  B C c be  three  rays  of  light  which  strike 
the  particle  ; the  ray  A a which  impinges  on  the  top 
of  the  particle  is  either  absorbed  or  directly  reflected 
back  ; the  ray  B 6,  slightly  to  one  side  of  A a,  is  reflected 
along  h E and  does  not  produce  an  effect  at  any  great 
distance  from  the  regular  image ; the  ray  C c which 
strikes  the  particle  considerably  to  one  side  is  reflected 
along  cD  and  into  the  glass  along  D H to  meet  the 
back  surface  of  the  glass  at  H.  At  H part  of  the 
light  is  refracted  and  part  reflected  along  H N,  entering 
the  film  again  and  producing  chemical  action. 

The  amount  of  liglit  reflected  depends  on  the  angle 
of  incidence,  increasing  rapidly  as  the  critical  angle 
(§  12)  is  approached,  and  afterwards  total  internal  re- 
flection takes  place. 

We  thus  see  that  when  the  ray  DH  has  a small 
angle  of  incidence,  very  little  light  is  reflected  and  no 
considerable  action  takes  place  at  points  near  to  the 
regular  image,  but  as  H recedes  from  K and  the  angle 
of  incidence  increases  up  to  the  critical  angle,  the 
amount  of  light  reflected  becomes  large,  a black  ring  is 
formed  round  the  regular  image. 

A remedy  proposed  for  the  prevention  of  halation,  is 
to  paint  the  back  of  the  plate  black,  but  it  is  hard  to 
see  how  this  can  help,  since  the  light  which  does  the 
harm  never  emerges.  It  would  be  much  better  to 
have  the  back  of  the  plate  ground  to  prevent  any 
regular  reflection.  Several  kinds  of  plates  are  now 
made  in  which  halation  is  prevented,  either  by  the 
thickness  of  the  film  or  by  an  opaque  layer  between  the 
film  and  the  glass,  either  of  which  prevents  light  from 
passing  through  to  the  glass. 


INDEX 


The  numbers  refer  to  the  sections. 


Abeiiration,  chromatic,  80 

of  thin  lens,  82 

of  thick  lens,  83 

correction  of,  84 

spherical,  65& 

for  two  lenses,  67 

minimum,  69 

Abney  on  experimental  test  of 
shutters,  109 

on  measurement  of  density, 

22 

on  halation,  157 

on  photographic  effect  of 

solar  spectrum,  20 

on  illumination  of  the  field, 

115 

Absorption,  22 
Achromatic  combination,  84 
Achromatism  of  lenses  not  in 
contact,  86 

tests  of,  108,  109,  115 

Amplitude  of  a wave  or  vibra- 
tion, 5 

Angle  of  cone  of  illumination, 
115 

incidence,  10 

reflection,  10 

refraction,  10 

sharpness,  59 

view,  59 

Aperture,  effective,  of  stops, 
115 

Arago,  test  experiment  on  nature 
of  light,  3 

Astigmatism,  75,  76 

measurement  of,  106 — 115 

Axes,  principal  and  secondary, 


Axis  of  lens,  31 

Baille-Lemaire,  rapid  test  of 
lens,  116 

Boys,  photograph  of  flying  rifle 
bullet,  6 

Calculation  of  angle  of  view  of 
lens,  59,  61 

of  chromatic  aberration, 

83,  84 

of  conjugate  foci,  31 

of  depth  of  field,  53 

of  distortion,  79 

of  efficiency  of  shutter, 

133 

of  elements  of  a combination 

of  lenses,  58 

of  enlarging  and  reducing 

tables,  151 

of  exposure.  121-122,  126, 

152 

on  moving  object, 

127,  128 

graphical,  39,  51 

of  magnification,  37 

of  principal  focal  length, 

33,  49 

of  spherical  aberration,  65c, 

66,  68 

of  size  of  plate  covered  by 

given  lens,  59 
Centering,  test  of.  111,  114 
Chromatic  aberration,  80 

of  thin  lens,  82 

of  thick  lens,  83 

Euler  on,  89 

Dollond  on,  89 


327 


328 


INDEX 


The  numbers  refer  to  the  sections. 


Chromatic  aberration,  Klengen- 
stierna  on,  89 

Newton  on,  89 

Circle  of  least  confusion,  75 
Clairaut,  combination  of  lenses, 
91 

Clerk-Maxwell,  electromagnetic 
theory  of  light,  4 
Colours,  photography  in,  23 

Lippman,  24 

Colson,  M.,  on  pinhole  photo- 
gi-aphy,  28 

Combination  of  lenses  in  contact, 
40,  91 

not  in  contact,  52 

indirect  method  of 

calculation,  92 
Condenser,  use  of,  141(X,  142 

construction  of,  143 

proper  position  of,  153 

Cone  of  illumination,  115 

outside  which  aperture 

begins  to  be  eclipsed,  59 

Dallmeyer’s  wide  angle  land- 
scape lens,  58 

telephotographic  lens,  60, 

124 

magnification  of, 

62 

Darwin,  Major  Leonard,  lens 
testing  at  Kew,  112 
Density,  measurement  of,  19 
Design  of  lenses  (chapter  iv),  91, 
105 

Determination  of  nodal  points, 
105 

principal  focal  surface,  106 

■ — • — depth  of  focus,  106 

astigmatism,  106 

• flat  field,  106 

distortion,  107,  115 

Deviation,  16 
Dispersion,  16 

irrationality  of,  81 

Distortion,  experimental  deter- 
mination of,  107,  115,  116 

calculation  of,  79 

— — cause  of,  78 


Distortion,  correction  of,  99 
— — due  to  diaphragm,  77 

due  to  wrong  position  of 

plate,  64 

Doliond,  correction  of  dispersion, 
89 

Efficiency  of  shutters,  131 
Emission  theory  of  light,  3 
Enlarging,  140 — 145 

Woodward’s  apparatus, 

141a 

optical  principles  of,  141 

modern  arrangements,  143 

by  daylight,  145 

with  a box,  149 

tables  for,  151 

Ether,  the,  7 

Euler  on  chromatic  aberration, 
89 

Exposures,  118 

with  telephotographic  lens, 

124 

on  objects  in  motion,  127, 

128 

duration  of,  with  shutter, 

130 

unequal  at  dilferent  parts 

of  plate,  137 

Faults  of  construction,  examina- 
tion for.  111 

Field,  curvature  of,  115,  116 

of  equal  brightness,  108 

depth  of,  156 

determination  of  flat,  106 

of  illumination  of,  115 

free  from  distortion,  107 

Flare  spot,  correction  of,  98  - 

test  for,  112 

Focal  length,  33 

of  thick  lens,  48 

determination  of,  41, 

102,  115,  116 

planes,  35 

plane  shutter,  138 

lines,  71,  72 

construction  for,  7Sa 

distance  between,  74 


INDEX 


329 


The  numbers  refer  to  the  sections. 


Focus,  33 

range  of,  42 

determination  of  depth  of, 

lf)6,  116,  155 
Focussing,  latitude  in,  90 
Frequency  of  vibration,  5 
Fresnel,  on  wave  theory  of  light,  4 

Geometrical  optics,  9 

construction  for  the  image, 

36 

constraints,  146 

practical  applications, 

147 

Graphical  calculations,  51 
Grubb,  measurement  of  focal 
length,  102 

Halation,  157 

Herschel,  combination  of  lenses, 
90 

Hertz,  electromagnetic  theory  of 
light,  4 

Hurter  and  Driffield  on  speed  of 
plates,  19,  126 

Illumination,  intensity  of,  13 

of  oblique  area,  lia 

cone  of,  59 

determination  of  field  of,  115 

of  plate,  118 

Incidence,  angle  of,  10 
Index  of  refraction,  10 
Irradiation  (or  halation),  157 
Irrationality  of  dispersion,  81 
Ives,  photography  in  colours,  23 

Jena  glass,  94 

Kew  Observatory,  system  of 
lens  testing  at,  112 — 115 
Klengenstierna,  on  correction  of 
dispersion,  89 

Langley,  visual  intensity  of  the 
solar  spectrum,  18 
Latitude  in  focussing,  90 
Law  of  reflection,  13 
refraction,  13 


Least  confusion,  circle  of,  75 
Lens  testing  (chapter  v),  101 

at  Kew  Observatory, 

112—115 

Lenses  forms  of,  30 

thin,  32 

thick,  43 

rapid  test  of,  116 

Light,  emission  theory  of,  3 

wave  theory  of,  4,  5,  6 

measurement  of,  13 

velocity  of  propagation  of,  6 

refraction  and  reflection  of, 

10 

unit  quantity  of, 

Lines,  focal,  71,  72 

construction  for, 

distance  between,  74 

Lippman’s  coloured  photographs, 
24 

Magnification,  37,  50 

with  telephotographic  lens, 

62 

Martin,  M.,  on  design  of  lenses, 
91,  105 

Measurement  of  density,  19 

astigmatism,  107 

distortion,  106,  116 

focal  length,  41,  102, 

105,  115,  116 

^ight,  13 

transparency,  108 

Moessard,  panoramic  photogra- 
phy,  65 

lens  testing  by  the  tourni- 
quet, 101,  103—109 
Monkhoven,  measurement  of 
focal  length,  102 

discussion  of  systems  of 

enlargement,  141^^ 

Newton,  Sir  Isaac,  on  theories  of 
light,  3 

experiment  with  prism, 

16 

on  correction  of  dis- 
persion, 81,  89 

Nodal  points,  determination  of, 
105 


330 


INDEX 


The  numbers  refer  to  the  sections. 


Nodal  points  of  thick  lens,  44 

of  a combination  of 

lenses,  53 

Numerical  examples.  See  Cal- 
culation. 

Oblique  pencils,  70 

central,  73 

Optical  centre  of  thick  lens,  44 
Optics,  physical  and  geometrical, 
9 

Orthochromatic  photography,  21 

Panoramic  photography,  65 
Pencils,  oblique,  70 

central,  73 

Perspective,  63 
Photography  in  colours,  23 
Photometry,  15 
Pinhole  photography,  26 
Prism,  16 

Rayleigh,  Lord,  on  pinhole  pho- 
tography, 28 

Reduction,  apparatus  for,  150 

tables  for,  151 

Reflection,  angle  of,  10 

law  of,  10 

total  internal,  12 

Refraction,  angle  of,  10 

law  of,  10 

through  two  media,  11 

index  of,  10 

at  spherical  surface,  31 

through  a thin  lens,  32 

Reversibility  of  optical  instru- 
ments, .57 

Rifle  bullet,  photograph  of  mov- 


Schott,  inventor  of  Jena  glass,  94 
Sensitometers,  126 
Sharp  image,  definition  of,  90 
Sharpness,  angle  of,  59 
Shutter,  diagram,  135 

focal  plane,  138 

Shutters,  129—139 

duration  of  exposure,  130 

efficiency  of,  131 


Shutters,  experimental  examina- 
tion of,  134 

Solar  spectrum,  Langley’s 
measurement  of  visual  inten- 
sity, 18 

photographic  effect  of, 

20 

Spectroscope,  17 
Spherical  surface,  refraction  at, 
31 

aberration,  65& 

calculation  of,  65c 

for  two  lenses,  67 

trigonometrical  me- 
thod, 68 

minimum,  69 

Standard  candle,  13 
Stewart,  panoramic  camera,  65 
Stops,  122—123 
Swing-back,  use  of,  64 

Telephotographic  lens,  Dallme- 
yer’s,  60 

exposure  with,  124 

angle  of  view  of,  61 

magnifying  power  of, 

62 

Test  of  achromatism,  108,  109 
Testing  of  lenses,  101 

at  Kew  Observatory, 

112—115 

with  the  tourniquet, 

101 

rapid,  116 

Total  internal  reflection,  12 
Tourniquet,  101,  103,  109 
Transparency,  125 

Yogel,  photography  in  colours, 
23 

Wallon,  100 

Warneke,  sensitometer,  126 
Wave,  theory  of  light,  4 — 6 
surface,  72 

Young,  Dr.  Thomas,  4 

Zeiss,  system  of  stops,  123 


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